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7.4 Permanent Component of Bias Temperature Instabilities

As previously mentioned, the recovery of a device subjected to BTI stress is the sum of a recoverable and permanent component. The recoverable component is assigned to oxide traps and can be explained by the four-state NMP model. In contrast, the permanent component is caused by defects with even larger time constants, and is mostly interpreted as the creation of interface states. Although the creation dynamics of the latter are still unclear, their presence is universally acknowledged [129, 130].

In several studies an increased number of interface states has been shown using several experimental techniques, such as ESR [131, 132], spin dependent recombination (SDR) [132], direct current voltage (DCIV) [133], charge pumping (CP) [134, 135, 136] and (math image) measurements [137]. To describe the permanent component, a purely empirical double well (DW) model was proposed and used in several studies [121, 108, 138]. However, the DW model only provides a rather phenomenological explanation of the experimental data. Recently, a new model relying on a hydrogen release mechanism has been proposed providing a physical based explanation for the accumulation of the permanent component of BTI [139, MWC7, MWC6]. Both models are discussed in the following.

7.4.1 Double Well Model

The \( \PbHDefect     \) configuration is a possible candidate for an interface trap contributing to the permanent threshold voltage shift. In its neutral state a Si atom located at the interface is saturated by a hydrogen atom. During stress, the hydrogen atom can be released from the Si atom and moves away, leading to a dangling bond at the interface, which is known as \( \Pb   \) center. The neutral and the charged state of the \( \Pb   \) center can be assigned to state \( 1 \) and state \( 2 \) shown in Figure 7.14 (left).

(-tikz- diagram)

Figure 7.14:  The double well model is based on a two state process. To account for bias dependent transition rates, a field dependent energy barrier is introduced. The strength of the field dependence is given by the prefactor \( \gamma     \), after [121].

The transition \( 1\rightarrow 2 \) describes defect creation and the opposite pathway the neutralization of the defect. The relations between the capture and emission time and the corresponding transition rates are given by

(7.31–7.32) \{begin}{align}     \tauc & = \frac {1}{\kAB } \quad \mathrm {and} \\ \taue & = \frac {1}{\kBA }.   \{end}{align}

Without considering a specified atomic configuration, the forward transition rate \( \kAB   \) and the backward transition rate \( \kBA   \) are modeled to be bias dependent by introducing a bias dependent energy barrier, see the configuration coordinate diagram shown in Figure 7.14 (right). The resulting barriers read [109]

(7.33–7.34) \{begin}{align}   \epsAB & = \VDWAB - \gamma F, \\ \epsBA & = \VDWAB - \VDWB + \gamma F \{end}{align}

with the \( F \) electric field and \( \gamma   \) a prefactor describing the bias dependence. The corresponding capture and emission times are then given by

(7.35–7.36) \{begin}{align}   \tauc & = \attemf \myexp ^{\beta \epsAB }, \\ \taue & = \attemf \myexp ^{\beta \epsBA } \{end}{align}

with \( \attemf    \) the attempt frequency. Using the stated relations various energy barriers can now be used to obtain a distribution of charge capture and emission times. Note that considering the energy levels \( \VDWAB   \) and \( \VDWB   \) as random independent numbers, no correlation between (math image) and (math image) would be obtained. However, the potential energy surfaces calculated from DFT simulations are a result of various forces acting on the atomic structure. As such a strong correlated between \( \VDWAB   \) and \( \VDWB    \) is present. Nonetheless, to simplify the model, the energy barriers \( \epsAB   \) and \( \epsBA    \) are assumed to be independently distributed. Although the double-well model provides not a very physics based explanation for interface states, its applicability is currently justified as it reproduces the experimental data well and allows easy implementation into device simulators.

7.4.2 Hydrogen Release Model

During the long puzzling history of BTI different mechanisms have been assumed to be responsible for the recoverable and permanent contributions [140, 141, 93, 142]. Recently a new model based on the interaction of point defects with interstitial hydrogen has been suggested to explain the permanent component of BTI. The model thereby allows to reproduce experimental data collected from ultra-long time experiments of more than eight month on a single device.

Conventionally, defects responsible for the permanent component have been associated with interface states which typically show very large time constants [93, 143]. Furthermore it has been suggested that, in addition interface states, hydrogen related donor traps can be created which could dominate the permanent degradation at longer times [139, 144, 145]. Thus instead of considering different contributors to the recoverable and permanent component it is rather more tempting to assume the same kind of atomic configuration of the traps responsible for both. The vital difference between traps contributing to the recoverable and permanent threshold voltage shift is due to their capture and emission time constants. In general, traps which have an emission time \( \taue   \) smaller that the experimental window \( \tRead     \), i.e. \( \taue < \tRead     \), obviously contribute to the recoverable \( \dVth   \), and traps with \( \taue > \tRead    \) are seen as the permanent \( \dVth   \). In the HR model a hydrogen atom can be released by a trap with \( \taue < \tRead     \), move through the oxide as interstitial hydrogen, and can subsequently be trapped by a defect with \( \taue > \tRead   \), thereby contributing to the permanent component.

The cornerstone of the HR model are recent DFT studies which have demonstrated that hydrogen can bind to Si – O – Si bridges in amorphous and defect free \( \SIO     \) [146, 147], as has been observed in previous experiments [145]. For this to happen, stretched Si – O – Si precursors have to be available, which is particularly likely close to the interface. It has to be noted that this stretched precursor is not available in crystalline (math image), where only a proton can bind in a stable manner to the bridging oxygen [148]. Considering the stretched Si – O – Si bond, the interstitial hydrogen can release its electron and bind to the oxygen. Alternatively, the hydrogen can break one of the Si – O bonds and form a hydroxyl group (– OH), which then faces the dangling bond of the other Si, leading to a configuration very similar to the \( E’ \) center [149]. Recently, it has been demonstrated that the energy barrier of this defect to become charged and discharged is in agreement with defects found to be responsible for the recoverable component in pMOSFETs [MWC16]. Furthermore, since the H can be removed over a thermal barrier, leaving an electrical inactive defect precursor behind, this defect appears to be consistent with the observed volatility of oxide traps [MWC12] as well, see Section 7.3.3.

The HR model is built around the hydrogen which can be released at the gate side during stress and in the following moves through the oxide towards the channel interface. As shown in the model in Figure 7.15, a defect available in a suitable precursor state provides a trapped hydrogen atom located at the gate site. When a stress bias is applied, the trap level of the trapped \( H^+ \) moves below the Fermi-level of the gate and thus the hydrogen is neutralized. After passing a thermal barrier, the neutralized hydrogen moves very fast towards the channel. Close to the channel, the neutralized hydrogen can get trapped by an empty \( H^+ \) trapping site, energetically aligned above the channel Fermi-level. As a consequence, an additional charge at the interface with very slow time constants is created, and thus contributes to the permanent component.

Nonetheless, it has to be mentioned that the microscopic trapping mechanism is very complex as it strongly depends on the atomic configuration of the involved defects. In such \( \SIO   \)/Si system defects can exhibit various atomic configurations which are mostly based on disorders of the perfect crystal structure due to nested hydrogen [150, 151, 152]. These anomalies differ in the displacement ant the bond angles between the single atoms thereby leading to different energy barriers for charge trapping. However, these energy barriers are an essential clue for the HR model as they primary define the time constants responsible for charge capture and emission. Hence defects responsible for the permanent threshold voltage shift require a large emission time, only structures exhibiting a large energy barrier for charge emission are of particular interest in context of the HR model. Finally, to provide validate the accurate description of the permanent contribution by the HR model, the energy barriers used within the HR model are in agreement with the energy barriers obtained from DFT simulations.

For the sake of completeness, it is worth mentioning that various configurations of the \( \SIO   \)/Si system involving hydrogen are available [150, 151, 152]. This makes the understanding of the detailed chemical trapping mechanism very complex.

The model discussed in the following relies on the HR mechanism, see Figure 7.15.

(-pstool- diagram)

Figure 7.15:  The schematic representation of the HR model shows the trapped \( H^+ \) at the gate side in the initial configuration (A). During stress the \( H^+ \) trap level is below the Fermi-level (B). Next, the \( H^+ \) is neutralized and can detrap via a thermal barrier and afterwards migrate very quickly towards the channel (C). There the neutralized \( H^+ \) becomes trapped by a precursor with its energy level above the Fermi-level of the channel (D). In particular at high temperatures, additional \( H^+ \) can be released from a reservoir at the gate side, [MWC6].

A simplified one-dimensional schematic for the HR model is shown in Figure 7.16.


Figure 7.16:  The simplified one-dimensional model shows the hydrogen which migrates through the oxide towards the channel. Thereby the hydrogen can get trapped, for instance bonded to an bridging oxygen atom, or detrapped, leading to interstitial hydrogen. Additional hydrogen can get released from a reservoir over a thermal barrier, [MWC6].

Since only a small number of hydrogen atoms are present in the gate stack, the model deals with absolute numbers and not with concentrations. The equation describing the temporal change of interstitial hydrogen at site \( i \) given reads

(7.37) \{begin}{align}    \PD {\Hz {i}}{t} = - \sum _j \kH \bigl (\Hz {i} - \Hz {j}\bigr ) - \sum _n T_{i,n} \{end}{align}

where the first sum runs over all neighboring sites \( j \), thereby describing the diffusion of the interstitial hydrogen which is considered a thermally activated hopping process with the rate

(7.38) \{begin}{align}   \kH = k_{ij} = k_0 \myexp ^{\frac {-q \EH }{\kB T}} \{end}{align}

and the prefactor \( k_0 = \DiffuH /a^2= 4\times 10^{10}\,\mathrm {s}^{-1} \) using the diffusion constant \( \DiffuH = 10^{-4}\,\mathrm {cm}^2/\mathrm {s} \) and the hopping distance of \( a = \SI {5}{\angstrom } \) according to literature [153, 154]. At each site \( i \) the interstitial hydrogen can get can trapped in its neutral configuration described by the trapping rates

(7.39) \{begin}{align}   T_{i,n} &= k_{01} \Hz {i} \bigl (\HTm - (\HzT {i,n} + \HPT {i,n}) \bigr ) - k_{10} \HzT {i,n}\ \label {e:Tin} \{end}{align}

via the rate \( k_{01} \) and the transition back \( k_{10} \). These rates are modeling using the Arrhenius law and read

(7.40–7.41) \{begin}{align}   k_{01} &= \frac {1}{\nu }\myexp ^{-\beta E_{01}} \\ k_{10} &= \frac {1}{\nu }\myexp ^{-\beta E_{10}} \{end}{align}

with \( \nu   \) the attempt frequency and the distributed energy barriers \( E_{01} \) and \( E_{10} \). At each site only a certain number of traps is allowed, which is considered by the term \( \HTm - (\HzT {i,n} + \HPT {i,n}) \) in the model. Next, the transition of the hydrogen from its neutral configuration either to the positively charged state or the transition back to interstitial hydrogen has to be considered. Both cases are described by the corresponding rate equations

(7.42–7.43) \{begin}{alignat}    \PD {\HzT {i}}{t} &=& -&k_{12} \HzT {i} + k_{21} \HPT {i} + T_i\\ \PD {\HPT {i}}{t} &=& &k_{12} \HzT {i} - k_{21} \HPT {i} \{end}{alignat}

while NMP theory is used to model the transition rates \( k_{12} \) and \( k_{21} \) from the neutral to the positively trapped state and vice versa.

Finally, close to the gate (either poly-silicon and/or metallization) a hydrogen reservoir is assumed to exists, consisted with NRA data [155, 156]. The interaction between the bonded hydrogen and the gate stack is described by

(7.44) \{begin}{alignat}   \PD {\HR }{t} &=& - &\kRz \HR + \kzR \Hz {i} \{end}{alignat}

for sites \( i \) at the gate/oxide interface, with \( \kRz     \) and \( \kzR    \) the forward and backward rates, respectively. An evaluation of the model against the recent experimental data is discussed in part three of this thesis.

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