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

5.3 Interpretation as distributed activation energies

An alternative interpretation of the temperature dependence of BTI stress and recovery can be given by applying models for chemical reactions involving distributed activation energies. Such reactions occur frequently for natural materials as e.g. fossil fuels, coals or natural gases and are therefore important in the petrochemical industry. These application fields lead to a good theoretical background for such reactions [Pri55; BB87] which is utilized here on BTI data. In fact, considering that the creation of charges close to the Si-SiO2 interface occurs mostly through the dissociation of atomic bonds [LD84; Fuj+03; Cam+06], a characterization from a chemical point of view seems very appropriate. In the following, the derivation of the equations needed to analyze the BTI data of Section 5.1 in this respect is presented. Naturally, this approach shows similarities to the temperature time approach of the previous section.

5.3.1 Derivation

A general assumption about the kinetics of processes which form a chemical transition is that they obey the differential equation [Pri55]

$(5.7) \begin{equation} -\frac {\tn {d}g}{\tn {d}t} = k g^n \end{equation}$

where $g$ is the concentration of the reactant of the kinetic process, $k$ the rate constant and $n$ the order of the reaction. For the time being, it is assumed that the processes which occur during the degradation of MOSFETs follow a first-order model ( $n=1$ ) [Ste00; Gra+11a; PG13b], keeping in mind that an expansion to higher orders remains possible. Also, if the concentration of reactants $g$ is not directly observable in the experiment, which is true within this thesis, the equations for a derived quantity as e.g. the (math image) are equivalent to the equations for $g$ [Pri55].

The rate constant $k$ is considered to follow Arrhenius’ equation leading to

$(5.8) \begin{equation} -\frac {\tn {d}g}{\tn {d}t} = \gls {k0} g \exp \left ( -\frac {E_\tn {A}}{k_\tn {B}T} \right ) \end{equation}$

which can be integrated for isothermal experiments to

$(5.9) \begin{equation} g = g_0 \exp \left ( - \gls {k0} t \e ^{-\frac {E_\tn {A}}{k_\tn {B}T}} \right ). \end{equation}$

$g_0$ is the value of $g$ at $\gls {t}=\SI {0}{\second }$ and (math image) the rate constant (also named frequency constant) with unit 1/s. The measurable quantity is the integral of the first order characteristic isothermal annealing function

$(5.10) \begin{equation} \Theta _1 = \exp \left ( - \gls {k0} t \e ^{-E_\tn {A}/k_\tn {B}T} \right ) \label {eq:charIsothermAnnealing} \end{equation}$

over all possible energy values

$(5.11) \begin{equation} G = \int _0^\infty g_0(E_\tn {A}) \Theta _1(E_\tn {A},t) \tn {d}E_\tn {A}. \label {eq:PrimakWithIsoTherm} \end{equation}$

The heart of this approach is the approximation of the characteristic isothermal annealing function (5.10) by a unit step function at $E_{\tn {A},0} = k_\tn {B}T\ln \left (\gls {k0}t\right )$ as shown in Fig. 5.20.

$ E_{\tn {A},0} = k_\tn {B}T\ln \left (\gls {k0}t\right ) $

The width of the transition in Fig. 5.20 is $4\times k_\tn {B}T$ or approximately 150 meV at 147 °C. That is to say, details in the activation energy spectrum smaller than a few $\gls {kB}\gls {T}$ are neglected. This will be verified later by suggesting particular activation energy spectra and comparing them to the results of simulated isothermal annealing data (cf. Fig. 5.21).

Replacing the isothermal annealing function with a unit step function allows solving (5.11) conveniently as

$(5.12–5.13) \{begin}{align} G &\cong \int _{E_{\tn {A},0}}^\infty g_0(E_\tn {A}) \tn {d}E_\tn {A} \\ \frac {\tn {d}G}{\tn {d}t} &\cong -g_0(E_{\tn {A},0}) \frac {\tn {d}E_{\tn {A},0}}{\tn {d}t} \{end}{align}$

and, since $E_{\tn {A},0} = k_\tn {B}T\ln \left (\gls {k0}t\right )$ ,

$(5.14) \begin{equation} g_0\left ( k_\tn {B}T \ln \left (\gls {k0}t\right ) \right ) \cong -\frac {t}{k_\tn {B}T} \frac {\tn {d}G}{\tn {d}t}. \end{equation}$

To verify this, Fig. 5.21 compares some selected activation energy distributions with the result of isothermal experiments analyzed with the Primak approach [Pri55].

Indeed, the approach does not resolve abrupt changes in the distribution of the activation energies accurately but catches the general aspects of the distributions. The method is only a valid approach to analyze a distribution of activation energies if the distribution is sufficiently broad.

However, a disadvantage of this approach is that the rate constant (math image) is undefined and must be chosen appropriately. The only way to determine is by comparing isothermal experiments at different temperatures [Pri55]. The traces in the energy spectrum need to overlap for correct rate constant values. This can be used to determine the unknown rate constant from isothermal experiments. Also the opposite is true: if different rate constants for different processes are existent isochronal experiments at different temperatures exhibit different energy distributions. In other words, the effect of the frequency factor is to shift the energy distribution along the activation energy axis in an isothermal experiment [Pri55].

In solid state reactions the frequency factor can be dependent on the temperature as $\gls {k0}\propto T^m$ with $m=-1.5\dots 2.5$ [CL08]. This means that in a (math image) range far away from the absolute zero temperature point, either increases or decreases with temperature according to the power law coefficient. This does not include a distribution of [BB87] as it is expected for BTI defects from TDDS studies where the individual defects show very different minimum time constants [Gra+10a].

5.3.2 Application to BTI data

The Primak approach is well applicable to NBTI data as shown in Fig. 5.22, when only (math image) is varied and and are always kept at the same value.

$ \gls {dVth}\left (\gls {tstr}\right ) $

The rate constant (math image) is chosen to be 10⁹/s such that a good overlap is given in the high energy region of the plot. However, for an improved matching in the low energy region below about 1 eV a lower rate constant of about 10⁶/s leads to better results (not shown). This indicates already that, despite the fact that an average value may also capture the behavior approximately, the reactions occurring during NBTS do experience different frequency constants.

Considering only the high energy tail of the activation energy distribution by measuring at the maximum achievable $\gls {Tstr}=\SI {430}{\celsius }$ , while keeping (math image) at 30 °C, the maximum of the spectrum can be unambiguously measured. See Fig. 5.23 for the $\gls {dVth}(\gls {tstr})$ data and Fig. 5.24 for the energy spectrum.

The characteristics in Fig. 5.23 and Fig. 5.24 are measured at different readout voltages (−2.0 V and −6.5 V) to account for the large shift of the (math image) of up to 4.5 V. This is necessary because at −2.0 V readout bias the shift is as large that the drain current becomes too small to be measurable. Increasing the readout bias to −6.5 V allows measuring also such large drifts and does not considerably change the parameters of the distribution, see the two characteristics at −9 V stress bias. The exact impact of the recovery bias is shown in Fig. 5.25 by comparing a virgin versus a stressed transfer characteristics.

The shift is fairly horizontal but includes a decrease of the sub-threshold slope which is usually attributed to an increase of the interface trap density [Sch06; ANG09b]. That is to say, the (math image) value might be altered by more or less interface traps but still reflects the relative number of NBTS created charges, as can also be seen in the $\gls {Vstr}=\SI {-9}{\volt }$ characteristics with different recovery biases in Fig. 5.23. Also, the −6.5 V recovery bias occurs only at the much lower recovery temperature which prevents any further degradation. Furthermore, also the impact of a zero gate and drain bias bake phase on the stressed device is shown in Fig. 5.25. The bake phase allows to return very close to the virgin state of the device [Kat08; BOG08].

The activation energy maximum is visible in the $\gls {dVth}(\gls {tstr})$ data as a slight saturation. The activation energy spectrum is in principle the derivative of the $\gls {dVth}(\gls {tstr})$ data. The point of inflection in the $\gls {dVth}(\gls {tstr})$ plot marks the maximum in the activation energy spectrum. Having a sufficient amount of data points near the maximum it is a rather facile task to fit a distribution of activation energies to the spectrum. The energy spectra resemble normal distributions. This is consistent with results from ESR measurements which propose the dissociation energy of, e.g. Si–H, to be normally distributed due to the amorphous structure of the SiO2 [Ste00]. The particular values of the mean activation energies could be erroneous since they depend largely on the frequency constant (math image) , which in turn can only be approximately evaluated by matching the vertical overlap of the data of the energy spectra plots. The dependence of the parameters on the stress bias is given in Fig. 5.33 in Section 5.5.

A particularity of the presented approach is that it is important to provide BTI data where only the (math image) temperature is varied and the recovery is kept always at the same level. This is because the product of chemical reactions is usually directly observable during the experiment. This would match most closely to on-the-fly (OTF) measurements for BTI research [Den+04a]. However, OTF measurements are seriously affected by errors due to the change of the mobility [Gra+08]. Furthermore, a transfer characteristic of the virgin device at the stress temperature is needed for the OTF approach. But at temperatures as high as 430 °C the measurement of the transfer characteristic itself degrades the device already considerably. Therefore, it would not be possible to obtain non-erroneous data from OTF experiments at high temperatures and therefore MSM data is favored.