#### Which Memristor Theory is Best for Relating Devices Properties to Memristive Function?

##### Ella M. Gale, Benjamin de Lacy Costello, Andrew Adamatzky

There are three theoretical models which purport to relate experimentally-measurable or fabrication-controllable device properties to the memristor's operation: 1. Strukov et al's phenomenological model; 2. Georgiou et al's Bernoulli rewrite of that phenomenological model; 3. Gale's memory-conservation model. They differ in their prediction of the effect on memristance of changing the electrode size and factors that affect the hysteresis. Using a batch of TiO$_2$ sol-gel memristors fabricated with different top electrode widths we test and compare these three theories. It was found that, contrary to model 2's prediction, the `dimensionless lumped parameter', $\beta$, did not correlate to any measure of the hysteresis. Contrary to model 1, memristance was found to be dependent on the three spatial dimensions of the TiO$_2$ layer, as was predicted by model 3. Model 3 was found to fit the change in resistance value with electrode size. Simulations using model 3 and experimentally derived values for contact resistance gave hysteresis values that were linearly related to (and only one order of magnitude out) from the experimentally-measured values. Memristor hysteresis was found to be related to the ON state resistance and thus the electrode size (as those two are related). These results offer a verification of the memory-conservation theory of memristance and its association of the vacancy magnetic flux with the missing magnetic flux in memristor theory. This is the first paper to experimentally test various theories pertaining to the operation of memristor devices.

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