Flash Temperature Testing and Modeling

Time to Bit Error Rate

When comparing to Arrhenius, we are looking for time to failure as a function of temperature. We can extract this from the data we have in figure 4.

Time to bit error rate measurement from ber data.
Fig. 6 Time to bit error rate measurement from ber data.The time to ber = 5e-5 is measured from the 40C, 60C and 70C curves.

Figure 6 above repeats the data of figure 4, and includes the extraction of the time to a bit error rate of 5e-5. Remember, for Arrhenius, the choice of ber is arbitrary. The horizontal green overlay line indicates the 5e-5 ber. We can extract the time when the 40C curve crosses 5e-5 as the reference point — thus it has a time to failure (TTF) ratio of 1.  You can see from the figure that 60C crosses at a TTF of 1.3 time that of 40C, and 70C crosses at a TTF of 2.2 times that of 40C. The 1.1eV Arrhenius model computes the 60C/40C TTF ratio as 3.0 and the 70C/40C ratio as 38.0. Arrhenius vastly overestimates the temperature acceleration factor between 70C and 40C. It is 2.3x too large at 60C/40C, and 17x too high at 70C/40C.

Hopefully at this point your faith in the accuracy of the Arrhenius model is beginning to waver. Fear not, we’ll put a stake though it shortly.

Bit Error Rate vs. Temperature

Now let’s add more data to the mix, and see if these trends are general.

ber model vs temperature at 3,000PE and 200H.
Fig. 7 ber model vs temperature at 3,000PE and 200H. The x axis is the temperature in C. The y axis the log of the bit error rate.

Figure 7 above shows aggregate data of the bit error rate vs. temperature at 3,000 PE cycles and 200 Hours. Data from multiple devices is included here – each data point is marked. The observed form is clearly super exponential. The parametric fit is:

$$ber \left( T \right) =
\alpha e^{ \left( {\beta \left( {T-\delta} \right) } \right)^\gamma }$$
Eqn. 3
T is the temperature in K
\( \alpha \) is constant scale factor
\( \beta \) is an exponential scale factor
\( \gamma \) is the power
\( \delta \) is a temperature offset – an activation energy?

I can’t give you a physical explanation of the functional form at this time, however it fits the data very nicely. For the data here, the fit parameters are:

\( \alpha = 4,176\)
\( \beta = 5.7e-3 \)
\( \gamma = 4.16 \)
\( \delta = 252 \)

Keep these parameters in mind – for the rest of the data sets I am going to use these parameters and not re-fit the data!

Bit Error Rate Temperature Model

We have observed that the temperature scaling law for the bit error rate is super exponential in temperature (eqn. 3). Next we will examine the ratios of bit error rates at different temperatures. Each data point here is a ratio of ber at two temperatures at a common data age and a a constant read rate.

I started with a sample set of data ages from 30 to 275 hours. I then computed the average bit error rate at each age from the measured data. Then, compute the ratio of each ber to the 100C value. I used multiple devices, 8 data ages each and 6 temperatures, so the data set is significantly larger than that for figure 7.

Bit error rate ratios to 100C
Fig. 8 Bit error rate ratios to 100C. The x axis is the second temperature in C. The y axis is the ratio of the bit error rate at the second temperature to the bit error at 100C.

Figure 8 above shows the bit error rate ratio scaling vs. the 100C bit error rate at the same age and cycle count. There are many more data points here as I have plotted the ratios for each age separately. You can see the 100C point clusters around a ratio of 1, which it should. The solid line is the model fit, based on eqn. 3. The fit here uses the parameters computed for figure 7 – I didn’t re-fit this data set. The parameters are:

\( \beta = 5.7e-3 \)
\( \gamma = 4.16 \)
\( \delta = 252 \)

Note that \( \alpha]\) drops out in the ratio. I find the fit to be remarkable, given that is is derives solely from the 200 hour data, and doesn’t include any of the other ages.

ber Temperature Scaling Law

The bit error rate temperature scaling law is the ratio of eqn. 3 at the two temperatures:

$$\frac {ber \left( T2 \right)}{ber \left( T1 \right)} =
e^{
\left( {
\beta ^ \gamma
\left( {
\left( {T2-\delta} \right) ^\gamma – \left( {T1-\delta} \right) ^\gamma
} \right)
}\right)}$$
Eqn. 4

 

Time to Failure Acceleration Behavior

The Arrhenius model computes a ratio of times to failure at different temperatures, and the failure criterion for SSDs is a particular bit error rate. As stated before, the Arrhenius acceleration factor can’t depend on the bit error rate chosen.

Eqn. 4 can be inverted when reading at a constant rate, allow us to compute an acceleration factor.

$$a_f = \left( \frac {ber \left( T2 \right)}{ber \left( T1 \right)} \right)^{\frac{1}{\left(k+g\right)}} =
\left( e^{
\left( {
\beta ^ \gamma
\left( {
\left( {T2-\delta} \right) ^\gamma – \left( {T1-\delta} \right) ^\gamma
} \right)
}\right)}
\right)^{\frac{1}{\left(k+g\right)}}$$
Eqn. 5

So, the acceleration faction isn’t Arrhenius at all.

Now, just to finally put this to rest, let’s plot the time to failure ratios and compare with the expected Arrhenius 1.1eV model. (By now you should be expecting the outcome…)

Time to ber = 1e-4 ratio vs temperature
Fig. 9 Time to ber = 1e-4 ratio vs temperature. The x axis the temperature in C. The y axis is the ratio of the time to reach a ber of 1e-4 at 3,000 PE cycles.

Figure 9 above plots the ratio of time to a 1e-4 bit error rate vs. temperature, with a 100C reference temperature. Again, the values around 100C cluster around 1. The model fit here still uses the parameters derived from figure 7 — I haven’t refit the data. Again, the fit is remarkable, and seems to indicate that the model is robust (if not physically understood).

The purple dashed line shows the Arrhenius 1.1eV acceleration factor ratio, which is 1 at 100,C, and is rapidly incorrect at lower temperatures.

The Blue dashed line is the “best” Arrhenius fit to the data, which has an 0.58eV effective activation energy. However, it is what I call a broken watch fit — it is correct only at 2 points! So it isn’t a match to the data either.

Perhaps we should call it the Arrh-ain’t-ius model?

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