1.
Introduction
After the original model for memristors proposed by Strukov et al.[1], some other important proposals have appeared[2–4]. Those former analyses describe distinct aspects related to their performance: current or voltage control, the nature of the current-voltage relationship, accuracy and even the interplay with different window functions, which were introduced mainly to include non-linear effects on ionic drift[4]. Additionally, adequacy for circuit modeling within SPICE? has been also discussed. Most of these models are fitted to experimental results by performing calculations which are sometimes rather complex, requiring a high computing capability. Some parameters are not even ascribed to any physical variable, merely fulfilling a mathematical need. In this work we wish to get a clear picture of the memristive structure performance, based on easily understandable parameters like those introduced by Strukov et al.[1]. Moreover, calculation of any variable should be straightforward and obtained from the digitized set of experimental results. This is the reason for using the results obtained in the first part of this work[5] in the case of a square waveform. They are further analyzed under the view of Strukov’s non-linear proposal. Several microscopic variables are determined from measurable quantities. These variables include the size of the doped region w, its maximum value wmax (and consequently the state variable x = w/wmax), and the product ημD which describes both the magnitude of the ionic species mobility under the external electric field and the direction of the filament-length change. Also, the velocity of the boundary between the doped and undoped regions can be determined. We estimate the resistance of the different regions and the voltages across them, which lead to the average electric field on each. The power being dissipated along each region and its time evolution are also calculated. An additional feature subject to wide discussion in the literature is the window function, which is designed to produce a decrease of the boundary velocity near the extreme values of the state variable. Instead of proposing it, we determine the experimental window function in our structures. A simple spreadsheet method for all parameter calculations is used. All calculations proceed from the experimental data starting from a plausible image of our structures.
The starting point for the analysis of our previous results is the information about current, voltage and memristance versus time data obtained by the authors in Ref. [5] for memristive structures based on thermally oxidized Ti films, which is represented in Fig. 1.
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Figure1.
(Color online) Data obtained from Chua’s analysis of the experimental data in Ref. [5]. This is the starting point for the analysis in this work.
2.
Our data and the model from Strukov et al.
The model of Strukov et al.[1] is related to the variables depicted in Fig. 2, where a schematic cross-section of a filamentary region which changes its size between w = 0 and w = wmax is shown. An intermediate value w is also indicated. In this model, positively ionized oxygen vacancies constitute the dopant within the conductive region of the titanium oxide thin layer. These vacancies constitute electron donors, but they also form paths for oxygen ion drift. According to the literature, switching models of metal oxide films can be based on either cationic or anionic transport through different defects. For cationic related mechanisms, active metals like Ag or Cu in the electrodes would perform better; anionic mechanisms, on the other hand, are related to oxygen ions and vacancies[6]. These last results were proposed in, for instance, the cases of TiOx[7, 8], HfOx[9], SiOx[10] and others.
The resistance values Roff and Ron correspond to w = 0 and to w = wmax respectively.
wmax denotes the size of the region where the boundary between the doped and undoped zones is moved back and forth under the effect of a periodic bias signal. The two basic expressions for this model are:
$Mleft( w ight) = {R_{{ m{on}}}}left( {frac{w}{{{w_{max }}}}} ight) + {R_{{ m{off}}}}left( {1 - frac{w}{{{w_{max }}}}} ight),$  | (1) |
$frac{{{ m{d}}w}}{{{ m{d}}t}} = {v_{ m{d}}} = frac{{{eta _{ m{d}}}{R_{{ m{on}}}}}}{{{w_{max }}}}ileft( t ight),$  | (2) |
where M(w) is the memristance as a function of w, wmax, Ron and Roff. In both expressions we exchanged the formerly proposed total thickness d of the film for the maximum filament length wmax. Eq. (2) states the boundary drift speed vd in terms of Ron, wmax, η = ±1 (the sign depends on whether the boundary between the doped and the undoped regions moves in the same direction as the current or not), the mobility μd associated to the doping defect in the film and the current i(t).
2.1
The ημd product
By differentiating Eq. (1) with respect to time and then combining with Eq. (2), we obtain:
$eta {mu _{ m{d}}} = frac{{w_{max }^2}}{{{R_{{ m{on}}}}left( {{R_{{ m{on}}}} - {R_{{ m{off}}}}} ight)}}frac{{{ m d}M/{ m d}t}}{{ileft( t ight)}}.$  | (3) |
Important resistance values are determined from the memristance curve by inspection: RonN = 2.36 kΩ, RoffN = 271.26 kΩ (minimum and maximum values, respectively) for the negative-voltage branch and RonP = 3.09 kΩ, RoffP = 141.98 kΩ for the positive-voltage branch. The initial resistance (that determined at t = 0) is R0 = 8.5 kΩ. Herein the actual asymmetry that the device exhibits is considered. An appropriate value for wmax should be calculated. We can assume that the resistance of the doped region is much less than that of the undoped region; the approximate width of the initial doped region in Fig. 1 is dd = 5 nm according to the SIMS study in Ref. [5], while d = 35 nm. Since Roff and Ron are respectively proportional to d ? dd and d ? (dd + wmax), an estimate for wmax ≈ 29.7 nm is produced. The time derivative of the memristance in Eq. (3) was numerically calculated. The plot of the product η μd including the corresponding sign of η is presented in Fig. 3.
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Figure3.
(Color online) The product ημd on a broken vertical axis with a linear scale in the lower part, plotted as a function of time. The break was introduced to emphasize the details around the zero crossings. The step at the middle of the cycle (17 s) results from the numerical calculation of the derivative used in Eq. (3).
Since the mobility has positive sign, the negative sign of this product at certain time periods must be ascribed to the parameter η. In Fig. 3, four time-intervals can be envisaged as indicated: period I extends approximately from 0 to 7 s, period II from 7 s to 17 s, period III from 17 s to 27 s and period IV from 27 s to 34 s. The signs of the voltage and the current are always equal for a passive device like the memristor. From Fig. 3 and Eq. (2), the sign of η has the effects indicated in Table 1, implying in turn the direction of change of the conductive filament size. The sign of vd is interpreted in terms of the length change of the filament in Fig. 2: a positive value corresponds to an enlargement of the filament and a negative one to a shortening.
| Current (or Voltage) sign | η > 0 | η < 0 |
| Negative | vd < 0, Shortening Period II | vd > 0, Enlargement Most of Period I |
| Positive | vd > 0, Enlargement Period III | vd < 0, Shortening Period IV |
Table1.
The sign of the parameter η in Eq. (2), related to the boundary movement.
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| Current (or Voltage) sign | η > 0 | η < 0 |
| Negative | vd < 0, Shortening Period II | vd > 0, Enlargement Most of Period I |
| Positive | vd > 0, Enlargement Period III | vd < 0, Shortening Period IV |
Then a filament enlargement means that some oxygen negative ions in the boundary (filament tip) are pulled out from their corresponding bonds in the lattice and swept towards the anode (during the negative semi-cycle, this becomes the earthed terminal in Fig. 2, but during the positive semi-cycle, it is the left-side terminal). On the other hand, a shortening means that some oxygen ions are brought from somewhere else towards the boundary and get trapped in the existing vacancies, and so are recombined.
As seen in Table 1, in periods II and III the respective shortening and enlargement of the filament are consistent with the movement of positively charged ionic species (oxygen vacancies) proposed by Strukov et al.[1] under the corresponding voltages. This is not the case during period IV, nor during most of period I.
A more complex situation is observed in period I. The above-mentioned consistency with transport of positive vacancies occurs during a very short time interval as indicated in Fig. 3. There are two extra changes of sign during this period. This fact suggests that a null value of the product during this period does not represent ionic immobility, but rather that two sources for the drifting oxygen ions exist:
a) The first one is the cathodic side region, where a low population of oxygen ions prevails. Ions from this source would move through oxygen vacancies, favoring in turn a shortening of the filament and being consistent with Strukov’s proposal. Since vd and i(t) have the same direction in this case (earthed toward biased terminal), then the sign for η is positive.
b) The second one is the undoped region close to the filament tip. Oxygen ions are produced by either impact ionization or Zener effects in this region. This is possible under the effect of a very high electric field, which will later be shown to exist. Clearly, oxygen ions coming from this source will produce a filament enlargement because they also move towards the anode. In this case vd and i(t) have opposite direction, yielding a negative sign for η. These ions undergo a difficult drift, since this must happen mostly through interstitial sites.
The non-monotonic behavior of ημd during periods I and IV indicates changes of the predominance of the two ion sources. From Fig. 3 we can state that this phenomenon is important whenever the conductive filament exhibits most of its largest extension because oxygen ions are lacking throughout it.
During period IV, ionic movement corresponds to oxygen ions placed at interstitials during period I, which come back to recombine vacancies, slightly decreasing the filament length in turn.
2.2
The state variable x and the boundary velocity vd
The time dependence of the state variable x = w/wmax can be obtained by solving Eq. (1) for x and substituting in the experimental M(t) values. Calculation yields the results plotted in Fig. 4(a). It increases from about 0.98 up to 1 (net filament enlargement) during period I. Then, during period II it decreases down to 0 at the middle of the cycle. These two periods occur under the negative bias. In period III it grows again up to the maximum value, but now under the positive voltage, and finally decreases again down to 0.92, still under positive voltage. These observations are consistent with those discussed from Fig. 3.
In Fig. 4(a) |ημd| is plotted on a logarithmic scale to appreciate its actual magnitude. It achieves different maxima along each period: In II and III it reaches about 10?14 m2/(V·s) as x approaches 0, while in periods I and IV it reaches only about 10?16 m2/(V·s), as x is slightly below the maxima (x ~ 0.98, 0.92).
By substituting the ημd values in Eq. (2), the speed vd of the boundary can also be calculated. In the bottom part of Fig. 4, the boundary velocity vd and its absolute value are plotted to clearly show its sign and magnitude. Maximum values of about 17 nm/s are achieved during periods II and III (as x is close to 0). In periods I and IV the maximum velocity becomes only about 1 nm/s (as x is close to 1). The positive (negative) signs in the plot indicate the time intervals during which a filament enlargement (shortening) happens. A more accurate value of the zero-crossing times of vd is attained from the logarithmic plots.
The velocity behavior confirms the statements given from Fig. 3, including Table 1.
2.3
Partial memristances, voltages and electric fields
Other variables that can be assessed are: the memristances Md (doped region) and Mud (undoped region) corresponding to both terms in Eq. (1) and the voltages Vd, Vud and i(t)·Ron across the three regions connected in series (the doped filamentary section, the undoped convertible region and the series resistance represented by Ron). Also, the electric fields Ed and Eud in the doped and undoped regions as functions of time were calculated. These results are shown in Fig. 5.
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Figure5.
(Color online) (a) Partial memristances of the doped and undoped regions. (b) Voltage drops in each region considered and total voltage across the structure. (c) Magnitude of the average electric field strength across the doped and undoped regions, all of them as functions of time.
The memristance of the doped region follows a trend similar to that of the state-variable as expected, while that of the undoped region behaves contrarily. Note the very different resistance scales for each: the doped region varies from 1 up to about 3100 Ω and the undoped from less than 1 Ω up to more than 100 kΩ. In Fig. 5 (b), the voltage on the doped region is always the lowest, as expected. The voltage on the undoped region clearly grows due to the filament shortening during periods II and III. During these periods, large mobility values are reached. They are about two orders of magnitude larger than those in periods I and IV. Also, the largest variations of the partial memristances, filament length (w size) and drift speed of the boundary take place. The voltage drop on the series resistance becomes important as the filament approaches the maximum wmax. As expected, the addition of the three contributions yields the total voltage applied.
Special attention should be paid to the significantly large electric field strength on both (Fig. 5(c)) the doped and the undoped regions, bearing a momentary maximum at a certain time in every period, with values as high as 1.7 × 107 and 2 × 109 V/m, respectively. This difference of around 2 orders of magnitude is always maintained along the whole cycle.
Such high electric fields support the hypothesis of bond breakage by avalanche or tunneling effects, causing in turn the release of oxygen ions from the lattice. This process is easier if such ions are taken from sites where they are not completely bonded, like the tip of the filament. These oxygen ions under the negative bias should move towards the earthed terminal in Fig. 2, enlarging the doped region, but their movement would be forced to happen mainly through interstitial sites in the undoped region. This becomes a difficult drift process, leaving such oxygen ions at interstices not far from their original sites. The lower values of both mobility and boundary drift speed during this period as compared to those in periods II and III are consistent with this proposal. The small variations in the associated resistances of the doped and undoped regions also support this idea. This process would compete with, but dominate, the oxygen vacancies drift coming from the cathodic region under the negative bias during period I. The transition to period II, where shrinking of the filament is observed, takes place when a significant amount of cathodic oxygen ions can move from the left side in Fig. 2. This movement occurs through a region having many oxygen vacancies. Then, as soon as these vacancies start to recombine with oxygen ions, most probably at the filament tip, its size decreases down to 0 during this period. After finishing period II, the second half-cycle starts. Under the positive voltage during period III, the oxygen vacancy drift is reversed as compared to that during period II, enlarging again the conductive region until it reaches the maximum of 29.7 nm. Right away, period IV starts, and the oxygen ions formerly placed in interstitial sites during period I are brought back to occupy some oxygen vacancies, decreasing the filament size down to 27.32 nm. Effects of delay time associated to oxygen vacancy drift are not expected to be observed, since the frequency of the biasing signal is quite low. Besides, due to its very small mobility, the contribution to the total current of the vacancy species as a charge carrier is negligible.
2.4
Power calculations
An important variable regarding circuit design is the power dissipated during memristor performance. In Fig. 6 the power dissipated by the doped and undoped regions are depicted along with the total power through the structure as they change during the cycle time.
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Figure6.
(Color online) Time dependence of the instantaneous power dissipated by the doped and undoped regions. The total power is also plotted. Two local maxima in the total power exist, one for each semi-cycle.
Two maxima of the total power can be seen. The first one reaches 1.693 mW at the transition between periods I and II, due to maximum power dissipation in the doped region at this moment. Instead, the power on the undoped region becomes a minimum at such transition. The second one corresponds to 1.295 mW at the transition between periods III and IV, also due to a maximum dissipation by the doped region. Again, dissipation by the undoped region becomes a minimum. The undoped region dissipates at two maxima during each semi-cycle. There is one maximum during each period. Nevertheless, these maxima correspond to a lower value (~400 μW) than those for the doped region.
The time dependence of power can be correlated to the filament size by comparing it to the state-variable plot. This comparison is plotted in Fig. 7.
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Figure7.
(Color online) Time dependences of the state variable, the total instantaneous and the cumulative powers dissipated by the structure.
Clearly, the instantaneous maxima occur while the doped filamentary region acquires its largest length, once in every half-cycle. At these times the memristance reaches minima and the current reaches maxima. The average cumulative power dissipated is about 800 μW along the whole period.
2.5
Window function
Our discussion now takes on the concept of the so-called window function incorporated into Eq. (2) by Joglekar, Biolek, and some other authors[11–13] to avoid malfunctioning of the model. So far, the role of such functions has been mainly to decrease the speed of the boundary as it reaches positions close to the corresponding extremes of the state variable (x = 0, 1), adding in that way some non-linearity to ionic drift. If we interpret our experimental results in terms of some window function f(x), then the plot of our already calculated ημd product versus x must be analyzed to ascribe the x dependence of such a product to the mentioned function. Then, it will be substituted by ημd0 f(x), where the mobility μd0 does not depend on x. Eq. (2) becomes:
$frac{{{ m d}w}}{{{ m d}t}} = {v_{ m{d}}} = frac{{{eta _{{ m{d}}0}}{R_{{ m{on}}}}}}{{{w_{max }}}}ileft( t ight)fleft( x ight).$  | (4) |
The plot for the forthcoming analysis is given in Fig. 8.
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Figure8.
(Color online) The product ημd as a function of the state variable x = w/wmax. Breaks in the axes are introduced to emphasize detailed features close to x = 1 on the graph. Note the different scales in both horizontal and vertical axes and the magnification of the tip in the neighborhood of x = 1, which is showed in the inset.
If the variation of ημd observed in Fig. 8 is exclusively ascribed to some window function, then such a function looks quite different from those formerly proposed by different authors. Those functions tend to zero at points close to both extreme values of the state-variable x (0 and 1), always exhibiting a negative concavity. In our case, the curves show a different behavior. We found a negative concavity only in a very small region close to the maximum wmax (x = 1) as seen in the inset, where the mobility becomes 0 and reverts its sign. In this small region we fitted a cubic polynomial A0 + A1x + A2x2 + A3x3 for each period, which can change the sign of its concavity about x = 1. More importantly, no decrease is exhibited close to x = 0 where mobility in our result acquires its maximum value. A quadratic polynomial B0 + B1x + B2x2 was fitted to most of each period, whenever x is not very close to 1. The corresponding coefficients are presented in Table 2 for every period.
| Coefficient | Period I | Period II | Period III | Period IV |
| B0 | ?3.4582 × 10?14 | 3.27856 × 10?14 | 1.13749 × 10?14 | ?1.39974 × 10?14 |
| B1 | 6.8698 × 10?14 | ?6.51576 × 10?14 | ?2.23865 × 10?14 | 2.75294 × 10?14 |
| B2 | ?3.41166 × 10?14 | 3.23726 × 10?14 | 1.10131 × 10?14 | ?1.35324 × 10?14 |
Table2.
Coefficients obtained from fitting a quadratic polynomial to our (η μd) versus x data during the periods already observed in Fig. 3. The polynomials are of the form ημd (x) = B0 + B1x + B2x2.
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| Coefficient | Period I | Period II | Period III | Period IV |
| B0 | ?3.4582 × 10?14 | 3.27856 × 10?14 | 1.13749 × 10?14 | ?1.39974 × 10?14 |
| B1 | 6.8698 × 10?14 | ?6.51576 × 10?14 | ?2.23865 × 10?14 | 2.75294 × 10?14 |
| B2 | ?3.41166 × 10?14 | 3.23726 × 10?14 | 1.10131 × 10?14 | ?1.35324 × 10?14 |
After considering the corresponding sign of η during each period, the quadratic and cubic polynomials were factorized into the following forms, so defining the observed window functions as quadratic and cubic binomials:
${mu _{ m{d}}} = {mu _{{ m{d}}0}}{left( {1 - x} ight)^2},$  | (5) |
${mu _{ m{d}}} = {mu _{{ m{d}}0}}{left( {1 - x} ight)^3},$  | (6) |
where the pre-factor μd0 was found to be as indicated in Table 3.
| Period | Type/x range | μd0 ($setlength{voffset}{0pt}$frac{{{ m m^2}}}{{ m {Vcdot s}}}$) |
| I | Quadratic (x ≤ 0.9969) | 3.4582 × 10?14 |
| Cubic (x ≥ 0.9999) | 1.59684 × 10?7 | |
| II | Quadratic (x ≤ 0.9638) | 3.27856 × 10?14 |
| Cubic (x ≥ 0.985) | 9.19605 × 10?13 | |
| III | Quadratic (x ≤ 0.9511) | 1.13749 × 10?14 |
| Cubic (x ≥ 0.9606) | 1.1308 × 10?13 | |
| IV | Quadratic (x ≤ 0.992) | 1.39974 × 10?14 |
| Cubic (x ≥ 0.9982) | 5.33136 × 10?10 |
Table3.
Ionic mobility pre-factor determined from a least square fitting to quadratic and cubic polynomials. These polynomials yield the window functions of the samples in this work.
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| Period | Type/x range | μd0 ($setlength{voffset}{0pt}$frac{{{ m m^2}}}{{ m {Vcdot s}}}$) |
| I | Quadratic (x ≤ 0.9969) | 3.4582 × 10?14 |
| Cubic (x ≥ 0.9999) | 1.59684 × 10?7 | |
| II | Quadratic (x ≤ 0.9638) | 3.27856 × 10?14 |
| Cubic (x ≥ 0.985) | 9.19605 × 10?13 | |
| III | Quadratic (x ≤ 0.9511) | 1.13749 × 10?14 |
| Cubic (x ≥ 0.9606) | 1.1308 × 10?13 | |
| IV | Quadratic (x ≤ 0.992) | 1.39974 × 10?14 |
| Cubic (x ≥ 0.9982) | 5.33136 × 10?10 |
The mobility pre-factor μd0 is much larger in x ranges where a cubic window function is valid, i.e. in a very small neighborhood of x = 1. Hence, we take as a representative value, the average of those values valid in the largest ranges of x, i.e. those obtained from the quadratic window functions:
$mu_{ m d0}= (2.32 pm 0.61) times 10 ^{- 14} ;left(displaystylefrac{{{ m m^2}}}{{ m {V!cdot s}}} ight).$  |
The window functions deduced from the experimental results imply that no limitation for boundary speeds close to x = 0 needs to be imposed. Moreover, its magnitude reaches a maximum at this value as already seen in Fig. 4. This means that voltage bias cannot significantly change doping in the initially doped region in Fig. 2 as x tends to zero. On the other hand, while x tends to unity close to the transitions between periods I–II and III–IV (Fig. 4 (a)), it is seen in Fig. 5 (c) that the electric field reaches a minimum in both the doped (~ 599 V/m) and the undoped (~ 2.71 × 104 V/m) regions. In the undoped region this field is not enough to promote additional growth of the doped filamentary region beyond wmax. This fact constitutes a self-limiting condition for additional filament growth and it can be considered as inherent to Eq. (1), as will be demonstrated in the following section.
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Figure4.
(Color online) (a) the magnitude of the ημd product and the normalized position of the boundary (state-variable) as functions of time. (b) velocity of the boundary on linear and logarithmic scales for better appreciating both its sign and magnitude. The +/? signs indicate those of the boundary velocity during each period.
2.6
Resistance Modulation Index
Kavehei et al.[14] introduced the concept of resistance modulation index (RMI)
ho _{{
m{M}}}} = displaystylefrac{M}{{{R_{{
m{off}}}}}}$
${ ho _{ m{M}}} = frac{M}{{{R_0}}}.$  | (7) |
The memristance expressions given in Ref. [14]
m{off}}}}sqrt {1 - displaystylefrac{{2{mu _{
m{d}}}{R_{{
m{on}}}}}}{{{R_{{
m{off}}}}w_{max }^2}}varphi left( t
ight)} }
ight)$
m{on}}}}sqrt {1 - displaystylefrac{{2eta left( {{R_{{
m{off}}}} - {R_{{
m{on}}}}}
ight){mu _{
m{d}}}{R_{{
m{on}}}}}}{{R_0^2w_{max }^2}}varphi left( t
ight)} }
ight)$
which implicitly define ρM are no longer valid here, since they were deduced by assuming an identity window function.
The initial resistance is defined as
m{on}}}}left( {displaystylefrac{{{w_0}}}{{{x_{max }}}}}
ight) +$
m{off}}}}left( {1 - displaystylefrac{{{w_0}}}{{{x_{max }}}}}
ight)$
The corresponding time dependence of the experimental resistance modulation index is that given in Fig. 1 (Bottom) after scaling M by the divisor Ro.
This normalized memristance exhibits the ability of the structure to change its resistance with respect to the initial value R0. A wider range of ρM(t) values means a better memristive structure. The exponential nature of ρM(t) during some parts of periods II (increasing) and III (decreasing) corresponds to that from the memristance, due to the proportionality between them.
An interesting feature of the RMI is its dependence on the state-variable x. Although the time dependence of memristance is not monotonic at all, a linear dependence on x holds during the whole cycle. This is easily seen if we rewrite Eq. (1) and divide it by R0 after substituting x = w/wmax. The result is:
$frac{{Mleft( x ight)}}{{{R_0}}} equiv { ho _M}left( x ight) = frac{{{R_{{ m{off}}}}}}{{{R_0}}} - frac{{{R_{{ m{off}}}} - {R_{{ m{on}}}}}}{{{R_0}}}x,$  | (8) |
where such linear dependence is clearly exhibited. The y-intercept
m{off}}}}}}{{{R_0}}}}
ight)$
m{off}}}} - {R_{{
m{on}}}}}}{{{R_0}}}}
ight)$
m{off}}}}}}{{{R_{{
m{on}}}}}}$
m{off}}}}}}{{{R_{{
m{on}}}}}}$
m{on}}}}}}{{{R_0}}}$
Experimentally, these assertions mean that better memristors will be obtained if the conductive filament in Fig. 2 is completely formed prior to the first measurement. Also, the active material should be highly resistive when undoped but highly conductive when doped. This requirement depends on the doping level of the specific active material used. Furthermore, the series resistance should be minimized by using good ohmic contacts. Only in the limit when Roff
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Figure2.
Schematic cross-section of a conducting filament, illustrating the region where the length of such a filament changes from w = 0 to w = wmax under some specific voltage waveform, so defining wmax as the maximum w value under the bias considered. When w = 0 the total resistance becomes the maximum Roff and when w = wmax, the resistance becomes Ron. The polarity is defined by the connections as indicated.
$ m QF = 1 - frac{{ m NMR - NRS}}{{ m NRS}},$  | (9) |
in such a way that the closer to 1 the QF, the better the memristor.
These last definitions allow an immediate calculation of the above parameters after finding the memristance curve (Fig. 1-Bottom), since R0, Ron and Roff are found from direct inspection. By substituting the definitions of NMR and NRS, Eq. (9) is transformed to:
${ m QF} = 2 - frac{1}{{1 - frac{{{R_{{ m{on}}}}}}{{{R_{{ m{off}}}}}}}},$  | (10) |
where only Ron and Roff appear, making clear their importance for this calculation. This happens despite QF has been defined in terms of NMR and NRS. Furthermore, expression (10) simplifies to
m QF} approx 1 - {R_{{
m{on}}}}/{R_{{
m{off}}}}$
In the case of our study, the non-symmetric nature of the device under negative or positive bias yields two different straight lines, as seen in Fig. 9. For the negative bias we obtained a better memristive response than for the positive one in this experiment, since it encompasses a wider resistance change. The corresponding values of NMR and NRS are 31.91 and 31.62 for the negative bias branch (periods I & II), but only 16.70 and 16.33 for the positive bias branch (periods III & IV).
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Figure9.
Resistance modulation index as a function of the state variable.
The final QF as defined by the lines above is 0.9908 for the negative branch and 0.9773 for the positive one.
The minimum RMI, which corresponds to x = 1 (see inset in Fig. 9), is 0.28 (negative bias) and 0.36 (positive bias).
3.
Conclusion
The conjunction of Chua’s basic definition of memristors and a simple model for the internal features allows one to obtain deep insight into their performance. Although Strukov’s model is quite simple and easy to process mathematically, it provides a reasonable basis for analyzing and characterizing experimental devices, yielding a complete quantitative description of the internal behavior of the memristor structure, based only on spreadsheet calculations.
The window functions deduced from the experimental behavior of our structures are f(x) = (1 ? x)2 and g(x) = (1 ? x)3, which appear quite different from those formerly proposed by previous authors. These results imply that no speed limitation on the boundary exists close to x = 0, where the maximum ionic mobility μd (3.3 × 10?14 m2/(V s)) and boundary speed vd (1.9 × 10?8 m/s) were observed in our structures. This is due to the difficulty of, under the effect of bias, drastically changing the doping concentration of the doped region (size dd) which was originally built within the structure. At the other extreme case (x = 1), there is a self-limiting mechanism for additional growth of the filament, since the electric field is not strong enough to pull out oxygen ions from the lattice close to the filament tip.
The definitions of the Normalized Maximum Resistance NMR and the Normalized Resistance Span NRS introduced here, which in turn define the Resistance Modulation Index RMI, demonstrate the importance of distinguishing the initial value R0 from the extreme resistance values Ron and Roff. Better memristors correspond to the conditions R0 ≈ Ron and Roff
It would be worthwhile to test the data treatment outlined in this and the previous work as a characterization method for other memristive devices to check its usefulness on different cases.
Acknowledgments
The authors are grateful to CINVESTAV del I.P.N. for providing the resources to carry out this work.
