IEEE Circuits and Systems Magazine - Q2 2018 - 72

25

× 1026
Ndisc,0 = 25.0 . 10+26 m-3

Ndisc /m-3

20
15
10

Ndisc,0 = 12.0 . 10+26 m-3
Ndisc,0 = 0.06 . 10+26 m-3

5
0

0

10

5

15

20
t /s

25

30

35

40

Figure 32. Evidence for ac fading memory in the hafnium
oxide/titanium oxide bilayer memristor from aachen. memory state over time in response to the application of an
ac sine wave voltage of the form v applied = vt applied sin (2r f t)
with amplitude v applied = 1.2 V and frequency f = 1 Hz to
the top OE with the bottom aE grounded for initial states
N disc,0 ! " 0.06 · 10 26, 12.0 · 10 26, 25.0 · 10 26 , m -3 (note that the
voltage applied across the memristor nano-device of Fig. 29
is here simply computed as v m = v applied).

uImu = 0 A
uI m u = 1 A
uImu = 1.5 A
X-,0 X0,0

20

.
x /s

0

uImu = 3 A
uImu = 3.5 A
uImu = 4 A
X+,0

-20
uImu = 2 A
uImu = 2.5 A

-40
-6

-4

-2

0
x

2

4

6

x

Figure 33. drm of the bistable memristor with state equation
(52). the equilibria of the state equation (51) for each input
modulus value are clearly marked with circles filled with the
same colour chosen for the respective curve. the green-filled
circles are located at Xr -, 0 = - 1 - 2 = - 2.4142, Xr 0, 0 = 0, and
Xr +,0 = - 1 + 2 = 0.4142.

4
3
2
1
0
-1
-2
-3
-4
-5

x0 = 4
x0 = 2
x0 = 0.75
x0 = 0.25
x0 = -0.5
x0 = 0 (Separatrix)
x0 = -1.5
x0 = -2.5
x0 = -3.5
x0 = -5
0

0.5

1
t /s

1.5

2

Figure 34. state response to a dc positive stimulation of the
memristor modelled by the daE set (52)-(53) with current
modulus I m = 1 A.
72

IEEE cIrcuIts and systEms magazInE

V. Local Fading Memory and Bistability
The input-induced history erase effect is a manifestation
of the fading memory property in its original definition,
i.e. it leads to the appearance of a unique steady-state
dynamic behaviour under any possible initial condition
within the memristor state existence domain. However,
for the sake of completeness, there exists also a local
version of the fading memory capability, i.e., under a
given input, a nonlinear dynamic system-in particular a memristor-may admit two or more locally-stable
steady-state behaviours, each with a well-defined basin of attraction, within which, irrespective of the initial condition, a unique asymptotic behaviour emerges.
The local fading memory of multi-stable memristors has
been the object of a deep circuit- and system-theoretic
investigation in [47]. Here we wish to briefly review the
clear example presented in [47] to reveal the local fading
memory affecting memristors with multi-stability. Let us
consider the following DAE set, originally proposed by
L.O. Chua in [44]:
dx = f (x, i )
m
dt
3
= - x - 2x 2 + (3 + i 2m) x,

(51)

v m = R (x, i m) i m
= ^0.01x 2 i 2m h i m,

(52)

which falls into the class of first-order current-controlled
extended memristors [40]. Noting that the state evolution function is even with respect to the current, i.e.
f (x, i m) = f (x, - i m), the DRM of the memristor with state
equation (51) is reported in Fig. 33 under input current modulus values within the set I m ! {0, 1, 1.5, 2, 2.5, 3, 3.5, 4}.
Differently from all the cases discussed previously, here
each curve of the xo -x loci goes through the horizontal
axis a number of times, i.e. 3, under each constant positive or negative DC value for the input current I m . As a result, under each input current modulus the state equation
has three equilibria, specifically Xr -, I m (the leftmost one),
Xr 0, I m (the middle one, equal to 0 for all I m h, and Xr +, | I m |
(the rightmost one). By inspecting the directions of the arrows on the xo -x loci of Fig. 33 it is straightforward to realise that, under any DC stimulus with modulus I m , the
memristor state will approach the leftmost (rightmost)
equilibrium Xr -, I m ^ Xr +, I m h for any initial condition to the
left (right) of the unstable equilibrium Xr 0, I m = 0. In fact,
the two outer equilibria are locally-stable, and their basins
of attraction are separated by the unstable equilibrium located in the origin. With reference to Fig. 33, the equilibria
for I m = 0 A are clearly marked with green-filled circles on
the POP, which reveals the discrete non-volatile memory
capability of the memristor. Fig. 34 shows a DC simulation
result revealing the local fading memory the memristor is
sEcOnd quartEr 2018



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