IEEE Circuits and Systems Magazine - Q2 2018 - 67

driven with positive stimuli (compare the orders of magnitude for positive and negative inputs of equal modulus
in plots (a) and (b) of Fig. 23, respectively).

The xo -x loci of the physical 1M device from the University of Michigan-where Ua,n= 0.74 V-under negative DC stimuli are similar to the
family of curves shown in Fig. 27(b) for the purely-mathematical memristor model with Ua,n= 1.04 V.
14

sEcOnd quartEr 2018

Vm = 100 mV

.
x /s

1010

100

Vm = 250 mV
Vm = 500 mV
Vm = 750 mV
Vm = 1,500 mV Vm = 1,000 mV

-100

-10-10

-10-20

0.2

Vm = -1,500 mV

0

.
x /s

1) Absence of AC Fading Memory
in Wei Lu's Model with Ua,p = Ua,n
Interestingly, a single voltage-dependent parameter,
specifically the ion hopping barrier height U (v m), is
responsible for the change in the form of the state equation of the nano-device-see equation (32)-upon
the polarity of the stimulus. As a result, the physical
memristor nano-device from the University of Michigan-i.e. the one-port shown in Fig. 22 with R s = 0 X -
features asymmetric switching kinetics, and, consequently, is subject to AC fading memory. On the other
hand, the hypothetical nano-device, modelled by equations (32), (33), and (34) with U a, p = U a, n features symmetric on- and off-switching kinetics. This is evident
from a comparison between plots (a) and (b) of Fig. 27,
respectively showing the DRM of Wei Lu's model for the
hypothetical 1 M structure with U a, p = U a, n = 1.04 V -all
other parameters in equations (32), (33), and (34) are
set to the numerical values reported in Table III-under
a set of positive and negative DC values for the input
voltage, with modulus Vm assuming values within the
set {100, 250, 500, 750, 1000, 1500} mV (with reference to
Fig. 22, note that here v m / v m, M since R s = 0 X). As common in real-world bipolar resistance switching memories,
the state x evolves in only one direction under excitations with unique polarity, and the speed it features over
the course of its time evolution throughout the existence domain [0, 1] increases with the input magnitude,
irrespective of the stimulus sign. Particularly, under any
positive (negative) input voltage the state progressively
increases (decreases) towards its upper unitary (lower
null) bound with rate growing (shrinking) dramatically
as the state gets closer to (moves away from) the upper
bound. It follows that this purely-mathematical model
also features DC fading memory capability.
Remark 5: Remarkably14 , noting that the xo -x loci of
the real-world nano-scale device under positive DC
inputs are equivalent to the family of curves depicted
in plot (a) of Fig. 27 for the hypothetical memristor-
the two models share the same value for U a, p -the
huge acceleration the memory state experiences as
it approaches its upper bound under positive stimuli, i.e. under on switching, is in fact at the origin for
the need of a series resistor to limit the maximum
current flowing through the nano-device. All in all,

1020

0

0.4

x
(a)

0.6

0.8

1

0.8

1

Vm = -1,000 mV
Vm = -750 mV
Vm = -500 mV
Vm = -250 mV
Vm = -100 mV

0.2

0.4

0.6
(b)

Figure 27. drm of the Wei Lu model for the hypothetical 1m
structure with U a, p = U a, n = 1.04 V under a set of positive (a)
and negative (b) dc input voltages with modulus in the set expressed by Vm ! " 100, 250, 500, 750, 1000, 1500 , mV. the
symmetry in the switching kinetics of this hypothetical nanodevice descends from the equality g (x, v m) = - g (x, - v m).
With reference to plot (a), the state rate xo goes to infinity
at x = 1. under a given negative input-refer to plot (b)-the
state rate xo at x = 0 drops suddenly from the respective value indicated by a white-filled circle to 0 s -1, since the lower
state bound represents an equilibrium for the state equation
(33) under negative inputs.

exciting the physical memristor nano-device with a
positive DC voltage source without setting an upper
bound for the current flowing through it-known as
current compliance level-may lead to its irreversible
physical damage15.
Importantly, since the state evolution function of the
Wei Lu model for the hypothetical 1 M structure with
T
U a, p = U a, n is odd with respect to the voltage v m = Vm, M
falling across it-an attribute missing in the model of the
physical memristor device, i.e. the one-port of Fig. 22 with
R s = 0 X, since g (x, v m) ! - g (x, - v m) in equation (32) for
U a, p ! U a, n -the hypothetical memristor is not affected
15

As pointed out earlier, the Wei Lu's 1M device model is not well posed
at x = 1, where the state evolution function in the nonlinear differential equation (32) is unbounded. However, using a series resistor the
switching kinetics are much slower for state values in the second half of
the existence domain [0, 1], as may be evinced by comparing Figs. 23(a)
and 27(a), thus decelerating considerably the state in its travel towards
the upper bound under positive inputs.
IEEE cIrcuIts and systEms magazInE

67



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