IEEE Circuits and Systems Magazine - Q2 2018 - 50

(e.g. which input range and initial condition set would be
allowed for a well-behaved device response in each operating mode [36]), allowing a more conscious adoption of
the nano-device for circuit design, especially in view of
its promising application for the development of innovative mem-computing circuits [37], [38] and systems [39].
A recent circuit-theoretic analysis on the mathematical
model from Strachan et al. [17] has recently revealed the
emergence of a peculiar dynamic phenomenon in the tantalum oxide memristor fabricated at HP Labs. Despite the
nanodevice is endowed with analogue nonvolatile memory capability, i.e. it may store an infinite continuum of
states at power-off [40], under suitable periodic excitation
its past history is progressively erased, and, consequently, its steady-state behaviour is found to be unique. This
dynamic phenomenon, called history erase effect [33], is
well known to experts in nonlinear system theory, where
it is referred to as fading memory [41]. The input-induced
memory loss effect [42], initially discovered in numerical
investigations, was later observed in experiments on a
nano-device sample [33], further revealing the predictive
nature of Strachan's physics-based mathematical description, which is one of the most accurate memristor models
available in the literature. The history erase effect is not an
exclusive property of the tantalum oxide nanodevice from
HP Labs. This paper demonstrates that fading memory
emerges also in other real-world non-volatile memristors
([15], [16], [19]-[21]) irrespective of their materials and of
the physics laws governing their switching kinetics. The
significance of the capability of a memristor to "forget" its
history under appropriate stimulation lies in the fact that
a nonlinear dynamic system endowed with fading memory may admit a Volterra series-based mathematical representation [43]. Considerable research efforts are currently devoted to the development of simpler yet accurate
memristor mathematical descriptions [26], [27] for robust
numerical simulations [28]. In this respect, exploiting the
large body of knowledge available in the literature on the
Volterra series theory [43], promising results have been
recently achieved in preliminary investigations on the application of the powerful Volterra modelling paradigm to
classes of ideal and ideal generic memristor [44] circuits
[45]. The manifestation of the memory loss phenomenon
in real-world resistance switching memories [23] may allow the derivation of Volterra series representations for
generic and extended memristors [40] as well, paving the
way towards the development of a systematic approach
to memristor device modelling as well as to memristor
circuit and system design and optimization [46]. The paper is organized as outlined next. Section II provides an
overview on the latest memristor classification. Section
III presents a rigorous definition of the fading memory,
elucidates the reason why this dynamic phenomenon
50

IEEE cIrcuIts and systEms magazInE

may never emerge in ideal or ideal generic memristors,
and explains why for a generic memristor model, which
does not capture the physics of a real-world nano-device,
but is frequently used for exploring the potential of resistance switching memories in future electronics, any DC
input induces the emergence of fading memory effects,
while the response to AC periodic inputs may critically
depend upon the initial condition. Section IV, constituting the core of the manuscript, discusses the ubiquitous
nature of the input-induced memory loss phenomenon
through the circuit- and system-theoretic analysis of a
number of accurate physics-based mathematical models
of real-world resistance switching memories fabricated
in different technologies. Very importantly, the origin
for the appearance of history erase effects in real-world
non-volatile memristors is unveiled under both DC and
AC periodic excitations. Section V briefly touches upon
the emergence of a local form of fading memory in multistable memristors, whose asymptotic response to a given
DC or AC excitation would be the one, among a set of possible steady-state behaviours, whose basin of attraction
contains the initial condition2 . Conclusions are finally
drafted in Section VI.
II. Classification of Memristors
Defining a memory-resistor (memristor for short), the
1971 seminal paper [1] from L.O. Chua refers to a twoterminal circuit-theoretic element with a nonlinear constitutive relation of the form
h (q m, { m) = 0

(1)

between the device charge q m (t ) = # i m (t l ) dt l = q m, 0 +
-3
#0 3 i m (t l ) dt l, initially equal to q m,0 =T q m (0) = #- 03 i m (t l ) dt l ,
3
a n d t he de v ice f lu x { m (t) = # v m (t l ) dt l = { m, 0 +
3
T -3
#0 v m (t l ) dt l , with initial value { m,0 = { m (0) = #- 03 v m (t l ) dt l
(i m and v m respectively denote the current through
and the voltage across the memristor). The memory
capability of the memristor introduced in [1] is thus
evident: its electric behaviour at the present time
depends upon the past histories of the device v o l t a ge and current. According to the latest classifica tion available in [40], a two-terminal circuit element
u n ivocally described by equation (1), is re fe r re d
to as ideal memristor. If the constitutive relationship in equation (1) may be reca st a s q m = qt m ({ m)
^ { m = {t m (q m) h, where the charge (flux) is expressed as
a single-valued function of the flux (charge), the ideal
memristor is said to be voltage (current)-controlled,
3

2

The mechanisms at the origin of multi-stability in resistance switching
memories were theoretically investigated in [47], and the design of a
memristor circuit emulator with bistable behaviour under suitable DC
and AC stimuli was proposed in [48].
sEcOnd quartEr 2018



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