The MLC Bit Error Rate Equation

Update July 2014: modified bit-error equation

I have modified the MLC equation based on examination of further data. The result is a better fit, although there plenty of parameters! I am not claiming this is the final form, but it works quite well for all the data examined within the data ranges. I would suggest caution with extrapolations.

The empirically determined equation of the MLC NAND error rate as a function of data age and the number of reads, and P-E cycle count at a fixed temperature is shown as equation 1.

 $$E = hA^kR^g + \frac{1}{1+\left( \frac{b}{C}\right)^{d}}$$ Eqn. 1 C is the P-E cycle count A is the data age R is the number of reads since the block was written h is scaling term and is a function of C k is the aging exponent, and is function of C g is the read count exponent, and is a function of C b and d are constants

(We will get to the temperature dependence in a subsequent post.)
I have simplified equation 1 somewhat for clarity, since some of the parameters depend on the P-E cycle count. The first term shows there is power law dependence on the data age and read count (which is a measure of read disturb). This term expresses the growth in the error rate after a block is written and the data ages. h dep. ends on the cycle count as well. I am still trying to determine the form of the dependence. It is something like a power law, but may be a log-logistic (which is close to power law near the origin).

The second term is essentially the error rate at a given cycle count at a data age of 0, thus represents an offset. The form is log-logistic (an S-curve), which is close to a power law near the origin. Both terms have a steep dependence on the cycle count.

We see that the aging and read disturb effects are primarily power law. The read disturb effect has been reported as a power law previously.

Equation 1 was determined by fitting measured data sets. The first term comes from fitting aging data, and the second term comes from fitting cycling data at very short data age. I will show these fits in upcoming posts.

The g and k parameters both display dependence on the P-E cycle count. Interestingly, the exponents for age and read appear to have somewhat similar functional behavior, although the parameter values are quite different.I should emphasize that the function fits for these parameters will be less accurate, as they are second order fits. That is, fits to parameters which themselves are determined by fitting. Thus, we should view them with a greater degree of skepticism that the fit for equation 1. Nonetheless, it is informative to examine their behavior.

As shown in equations 2 and 3, they both have roughly a linear dependence on the P-E cycle count.

 $$g = g_a + g_b C$$ Eqn. 2

The parameter k seems linear as well, although the signs of gb and kb are different. That is, g decreases with cycle count, while k increases with cycle count.

 $$k = k_a + k_b C$$ Eqn. 3

A preliminary examination of h indicates that it is likely log-logistic. Perhaps it is reasonable for the cycle effect with age to be similar to the effect at short ages.

 $$h = \frac{h_a}{1+\left( \frac{h_b}{C} \right)^{h_d}}$$ Eqn. 4

So, in the end the equation E(A,C,R) at constant temperature looks like:

 $$E= \frac{h_a}{1+\left( \frac{h_b}{C} \right)^{h_d}} A^{k_a + k_b C} R^{g_a + g_b C} + \frac{1}{1+\left( \frac{b}{C}\right)^{d}}$$ Eqn. 5

I generally prefer to use equation 1 when describing the bit error rate since it is cleaner, and the cycle dependence in the parameters g and k is weak and the functional form determination is not as precise.

We’ll explore the temperature dependence in a future post, but the astute reader will notice that it is sufficiently complex that it needs to be measured.

Notes:
1. Mielke at al., “Bit Error Rate in NAND Flash Memories”, IEEE 46th Annual International Reliability Symposium Symposium, 2008.
Mielke at al., “Bit Error Rate in NAND Flash Memories”, IEEE 46th Annual International Reliability Symposium Symposium, 2008.