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Degradation of Electrical Parameters of Power Semiconductor Devices – Process Influences and Modeling

5.2 Interpretation using the temperature-time approach

In the following the results of the preceding Section concerning the temperature dependence of stress and recovery data of BTI are interpreted. The first approach presented in this Section is motivated by recent results on the BTI response of individual defects [Gra+10b]. This work showed that the temperature dependence of the capture and emission time constants of a single defect follow an Arrhenius equation [Gra+10a, Fig. 6&8]. Based on these findings it is possible to calculate how temperature changes accelerate or decelerate BTI stress and recovery. To facilitate presentation, an abstract temperature-time (math image) is introduced and applied to stress and recovery data.

5.2.1 Temperature dependence of single defects time constants

On MOSFETs with channel length and width in the sub-micrometer range the charge state of a single defect near the interface can have a measurable impact on the (math image). This is because following the definition of the capacitance \( CA=Q/V \) the impact of a single charge \( q \) on the (math image) is given by

(5.1) \begin{equation} \gls {dVth} = \frac {q}{\gls {Cox} \gls {A}}.   \end{equation}

If the area of the MOSFET (math image) is reduced, the impact of a single charge on (math image) increases. For an SiO2 thickness of 2.2 nm the area of the MOSFET needs to be smaller than about \(100x100\) nm to observe a (math image) induced by a single defect at the Si-SiO2 interface larger than 1 mV, which is around the precision of a conventional (math image) measurements [KU89; Rei+10].

The discharge of a single defect is visible as discrete steps in the \( \gls {dVth}(\gls {trec}) \) trace [Rei+10; TL+11c]. The height of this step depends on the position of the defect with respect to the dopants in the channel. The dopants determine regions inside the channel where the drain current flows [Bin+12]. A single defect which is situated at a position where the majority of the drain current flows impacts the drain current more than a defect which is far away from this high current density regions. Consequently, the step height in the recovery trace is a fingerprint of an individual defect. With this it becomes possible to investigate the charge emission of a single defect despite of other emission events of neighboring defects within the same device [Gra+10a]. The analysis of such an experiment is referred to as time dependent defect spectroscopy (TDDS) [Gra+10a].

Thorough analysis of recurring emission events of single defects after stress have revealed that the emission time (math image) at fixed recovery bias and temperature follows an exponential distribution as

(5.2) \begin{equation} p\left ( t_\tn {e} \right ) = \frac {\gls {te}}{\gls {taue}} \exp \left ( -\frac {\gls {te}}{\gls {taue}} \right ) \end{equation}

with a mean emission time constant (math image) (also named characteristic emission time) [Gra+10a; TL+11c]. By varying the temperature in those experiments it was found that the emission time constant changes with temperature following an Arrhenius equation [TL+11b; TL+11d; TL+11c; Rei+10; Gra+10a]

(5.3) \begin{equation} \gls {tau} = \gls {tau0} \exp \left ( \frac {E_\tn {A}}{\gls {kB}T} \right ).   \label {eq:tauArrhenius} \end{equation}

An equivalent argumentation applies for the capture time constants (math image) of BTI defects. In Fig. 5.9 the temperature dependence of the emission and capture time constants of a few selected defects in a single device are depicted.


Fig. 5.9: Single defect emission (left) and capture (right) time constant temperature dependence from [Gra+10a, Fig. 8]. The defects are labeled arbitrarily. The almost linear relationship of \( \log \gls {tau} \) over \( 1/T \) suggests an Arrhenius-like temperature activation as (5.3). The corresponding activation energy is written next to the particular characteristic.

The capture and emission time constants of all defects are well represented by an Arrhenius equation (5.3). This temperature activation indicates that the microscopic charge exchange is not an elastic tunneling process [Gra12]. It is rather consistent with a nonradiative multiphonon theory [HR50; Vui+89]. However, the activation energy values appear to be widely distributed in a range of 0.6 eV to 1.39 eV for the emission and 0.4 eV to 0.99 eV for the capture events, respectively. This suggests that the thousands of defects in a micrometer sized device follow an arbitrary distribution of activation energies. This idea will be treated in detail in Section 5.3.

For the sake of completeness, Fig. 5.10 shows the dependence of the capture time constants of individual defects on the stress bias.


Fig. 5.10: Single defect capture time constant dependence on the stress bias from [Gra+10a, Fig. 6]. The capture time constants of all defects decrease with increasing stress bias or temperature. A model for this decrease based on SRH theory fails to reproduce the data.

A decrease of the capture time constants with increasing stress bias is observed, which does not follow a simple model derived from SRH [SR52; Hal52; Pot+07] theory [Gra+10a]. The emission time constants decrease with decreasing bias as shown in Fig. 5.11, even though there exist also defects whose emission time constants do not change with bias at all [Gra+13a].


Fig. 5.11: Dependence of the emission time constants of two selected defects on the recovery or readout bias from [Gra+13a, Fig. 9]. The extension of the graph towards the depletion region was achieved using a dynamic TDDS approach [Gra+13a].

To conclude, TDDS studies on individual defects have univocally shown that the mean capture or emission time constants of the defects constituting BTI follow an Arrhenius equation. Using this Arrhenius equation the construction of an abstract temperature-time is possible which accounts for arbitrary distributions of time constants.

5.2.2 Derivation of the temperature-time

To derive a transformation of the time constants of the defects activated during NBTS it is understood that the activation energy (math image) does not change from one temperature \( T_1 \) to any other temperature \( T_2 \). It is remarked that this is only valid if entropy changes are neglected. By rearranging (5.3) for (math image) one obtains

(5.4) \begin{equation} E_\tn {A} = k_\tn {B}T \ln \left ( \frac {\gls {tau}}{\gls {tau0}} \right ) \end{equation}

and further from \( E_\tn {A}(T_1) = E_\tn {A}(T_2) \),

(5.5) \{begin}{align} \gls {tau}_2 &= \gls {tau0} \exp \left ( \frac {E_\tn {A}(T_1)}{k_\tn {B}T_2} \right ) \notag \\ &= \gls {tau0} \exp \left ( \frac {k_\tn {B}T_1\ln \left (\gls {tau}_1/\gls
{tau0}\right )}{k_\tn {B}T_2} \right ) \notag \\ &= \gls {tau0} \left ( \frac {\gls {tau}_1}{\gls {tau0}} \right )^{T_1/T_2}. \label {eq:tauT1T2} \{end}{align}

With this the change of the defect time constant \( \gls {tau}_1 \) at temperature \( T_1 \) to another time constant \( \gls {tau}_2 \) at a temperature \( T_2 \) can be calculated, provided \( \gls {tau}_0 \) is known. To handle arbitrary time constant distributions of many defects the temperature-time (math image) can be defined as [Pob+11b]

(5.6) \begin{equation} \vartheta \left (T\right ) = \gls {tau0} \left ( \frac {t}{\gls {tau0}} \right )^{T_\tn {meas}/T}, \label {eq:Tt} \end{equation}

which transforms BTI stress or recovery data measured at the temperature \( T_\tn {meas} \) to any other reference temperature \( T \). Providing the physical correctness of our assumption, this approach makes it possible to transform data from a fixed experimental window of about 10−6 s to 106 s to an extended time frame. For example, measuring at colder temperatures than the reference temperature rescales the time axis toward smaller values. This allows decelerating fast transitions to measure them with reasonable measurement intervals on conventional equipment. On the other hand, if \( T_\tn {meas} > T \), defects with time constants much larger than the experimental window may be analyzed in a short-time experiment [Pob+11b; PG13b].

All considerations that follow are based on the assumption that NBTI is the response of numerous defects which behave approximately as a first order process with the time constants given by (5.3). The applicability of this assumption has been demonstrated previously [HDP06; Hua10; Gra+11a; Pob+11b]. It is remarked that the assumption that (5.3) is valid does not imply an assumption on the microscopic nature of the defects constituting NBTI [PG13b].

It is further remarked that the presented approach allows for very precise determination of the lifetime of devices by accelerating the degradation with temperature. This is far more precise and accurate than the conventional stress voltage scaling approach [Pob+11b].

5.2.3 Application to recovery temperature dependence

The temperature-time approach is applied to the NBTI recovery data measured at different temperatures as shown in Fig. 5.12.


Fig. 5.12: NBTI recovery data from Fig. 5.6 at different (math image) interpreted with the temperature time and compared to a real-time measurement [PG13b]. The voltage and temperature sequence for the experiment is sketched in Fig. 5.5.

In order to obtain this plot an average minimal emission time constant (math image) for 0 eV activation energy is required to be determined by minimizing the vertical difference in the overlapping data. The (math image) value obtained in this way is in the range of the inverse phonon frequency and around 1013 Hz [TPZ72; Ber+06; Aic10; Gar+12].

By using the temperature-time approach and finding an appropriate value for \( \tau _0 \) one single universal recovery trace was observed. This recovery trace reproduces a two day long reference measurement at 100 °C chuck temperature, which proves the validity of the concept. It is remarked that the overlapping of the data could be improved by considering two independent distributions with different values for (math image) as done previously [Gra+11a]. However, considering a single effective distribution with a single (math image) appears to be sufficiently accurate for the present purposes [PG13b].

This result confirms that the temperature dependence of NBTI recovery can be understood by a simple collection of temperature activated first-order processes: At low temperatures the time constants become larger and thus a smaller number of defects are able to anneal. Consequently the number of remaining charges and thus the measured (math image) is large. With increasing temperature the time constants become shorter and most charges are emitted during the recovery measurement. In the limiting case, when the recovery temperature even exceeds the stress temperature, nearly all the defects which had been created during stress anneal already before the first measurement point after a 10 ms real-time delay, resulting in a very small (math image) [PG13b].

The universal recovery trace may also be obtained in a single measurement through temperature switches during recovery, as shown in Fig. 5.13.


Fig. 5.13: Recovery data of Fig. 5.8 with (math image) switches during recovery interpreted with the temperature-time (5.6) [PG13b5.7 for the time and voltage chronological sequence. Data from Fig. 5.12 as gray background.

The effect of the time transformation (5.6) is to shift the experimental data along the time axis to larger values with higher temperatures. This allows extending the recovery transients to otherwise impractically long times, as shown in Fig. 5.14.


Fig. 5.14: Comparison between temperature accelerated recovery data and real-time data measured at 50 °C chuck temperature [PG13b].

It is remarked that further acceleration of the recovery with higher temperatures can lead to degradation even at \( \gls {Vg}=\gls {Vth} \), especially in devices with thin gate oxides, which can lead to the appearance of erroneous plateaus in the recovery transient [Gra+11c]. For this reason, recovery acceleration appears mostly beneficial for moderate stress conditions where the stress temperature is considerably lower than the maximum allowed recovery temperature.

An investigation of the quasi-permanent component by temperature acceleration of the recovery following moderate NBTS illustrated in Fig. 5.15 shows that the remaining level of degradation is rather small in devices with SiO2 or silicon oxynitride (SiON) technology. Furthermore, the concept of temperature acceleration for recovery appears to work in a wide variety of technologies with similar values of (math image).


Fig. 5.15: Recovery after NBTS with switches to higher temperature for several technologies [PG13b]. The vertical axis is normalized for better illustration. \( \gls {tau0} = \SI {e-11}{\second } \) is suitable for both SiO2 and SiON oxides with different thicknesses.

This justifies the use of temperature switches during recovery for the measurement of NBTI recovery data to acquire experimental data sets on long timescales [PG13b].

5.2.4 Application to stress temperature dependence

In Fig. 5.16 the result of the application of the temperature-time on the NBTI data measured at different (math image) is shown.


Fig. 5.16: The data of Fig. 5.4 interpreted with the temperature time (5.6). (math image) was chosen arbitrarily such that the vertical overlap of all traces is minimized.

Provided (math image) is chosen appropriately, all characteristics align to one single curve. The impact of the particular choice of (math image) is illustrated in Fig. 5.17.


Fig. 5.17: Impact of the particular choice of (math image) on the horizontal position of a characteristic measured at a (math image) larger than the reference temperature.

The (math image) value shifts data measured at a temperature different from the reference temperature horizontally along the \( \gls {Tt}_\tn {str} \) axis. Provided the assumptions for the derivation of the temperature-time are correct the data measured at 160 °C have to match the data measured at 235 °C after a certain amount of time. In the present thesis the exact value of (math image) is determined by reducing the mean square of the sum of the vertical difference of two traces in the region of overlap.

The good matching of the data at different (math image) suggests already that the analysis using (5.6) is a valid approach to interpret the acceleration of BTI stress with temperature. However, the approach opens also a way to acquire large experimental data sets in a minimum amount of measurement time. In order to prove this, a comparison of \( T \) accelerated MSM data to real-time data is shown in Fig. 5.19. This comparison requires to use (5.6) also for the recovery phase, in order to properly account for the different recovery delay times at different chuck temperatures. The according time and voltage sequences for the experiment are illustrated in Fig. 5.18.


Fig. 5.18: Evolution of the device temperature and gate bias for the experimental data shown in Fig. 5.19.


Fig. 5.19: Comparison between a long real-time MSM experiment and a (math image) accelerated MSM experiment [PG13b]. The different recovery delay times at different (math image) are accounted for by using (5.6). Chronological sequence of (math image) and (math image) can be found in Fig. 5.18.

The real-time MSM experiment needs half a week and was performed on a different wafer of the same technology with a different measurement equipment [Gra+11a]. In contrast, the temperature accelerated MSM experiment needed approximately one hour. The real-time MSM experiment measured the degradation with a 100 µs delay, which decreases the (math image) value due to recovery. These 100 µs at 170 °C correspond to approximately 20 ms at 30 °C for the temperature accelerated measurement. Exactly this delay time was used for the temperature accelerated measurement.

The \( T \) accelerated data is in agreement with the real-time data and exhibits the same gradual change in the power law exponent as the real-time data. The \( T \) acceleration appears therefore to be applicable to extend the experimental accessible time range to acquire long-term experimental stress data sets in a short amount of time [PG13b]. It is remarked that the approach works on all investigated technologies.