IEEE Circuits and Systems Magazine - Q2 2018 - 61

In each of the two plots arrows indicate the evolution
of the trajectory point on the respective loci as time
goes by (in plot (b) arrows are also numbered for better clarity12).
Remark 4: It is worthy to observe that the two-tone
input voltage results in the formation of an interesting
pinched hysteresis "double loop" on the v m -i m plane at
steady-state.
B. The titanium Oxide Memristor
from Hewlett Packard Labs
Taking inspiration from the physics-based Simmons tunnel barrier model [59], Pickett et al. presented a mathematical description capturing the complex dynamics
emerging in a titanium oxide-based memristor [15].
The Pickett mathematical description13, implemented in
PSpice [61] and taken as reference DAE set in a comprehensive memristor model comparison study presented
in [62], characterises most of the post-electro-forming
D-long titanium oxide-based film enclosed between two
platinum electrodes as a highly-conductive layer with
fixed resistance R s -also referred to as memristor channel-and models the remaining part as a narrow tunnel
barrier of length w and voltage drop v g = v m - R s · i m
(see Fig. 16).
The following nonlinear ODE dictates the time evolution of the tunnel barrier length w, chosen as memristor
state and confined within the closed set [1.1, 1.9] nm
dw = f (w, i )
m
dt
= foff sinh c

im
m step (i m)
i off

im
m- w m
· exp c - exp c w - a off wc
b
wc
im
m step (- i m)
- fon sinh c
i on
im
m - w m,
· exp c - exp c a on - w wc
b
wc

(30)

where the physical significance of the constants foff, a off,
i off, b, w c, fon, a on, and i on is reported elsewhere in the literature [15].
The current through the device is modeled as a tunnelling junction current [59]

+

Platinum
w

D

Rs
TiO2

TiO2-x

Platinum

-
Figure 16. Physical structure of the titanium oxide-based
memristor from HP Labs after electro-forming. Except for a
narrow tunnel barrier of length w, most of the D-long oxide
film between the two platinum contacts is occupied by an
oxygen-deficient conductive channel.

i m = F (w, v m, i m)
= sign (v g) ·

jo A
" z I e -B
(Dw) 2

With reference to Fig. 15, it is instructive to note that the trajectory point
spends a considerable amount of time around the second (starting from the
left) intersection of the xo -x loci with the horizontal axis-refer to the inset
of plot (a), depicting a close-up view of the lower left part of the xo versus x
loci-or, equivalently, on the higher conductance branch of the inner loop of
the vm -im loci in plot (b).
13
Kolka et al. [60] have recently introduced an improved mathematical
model of the Pickett model, which resolved a port equation ambiguity
at the origin of convergence issues, numerical errors, and non-physical
state solutions during time-domain simulations.
sEcOnd quartEr 2018

- (z I + e v g ) e -B

zI + e | vg |

,,

(31)
where A is the average memristor channel area, while
j o = ^e 2rh h, with e = 1.60 · 10 -19 C representing the elementary electronic charge, and h = 6.63 · 10 -34 m 2 · kg · s -1
standing for the Planck constant. Equation (31) may be
interpreted as a non-standard Ohm's law relating memristor current i m, state x, and voltage v m in implicit
form. Moreover, in equation (31) Dw = w 2 - w 1, where
w 1 and w 2 are defined as
w 1 = 1 .2 m w ,
zo

w2 = w1 + wc 1 -

9.2 m
m,
3 z o + 4m - 2 e v g

with z o symbolising the mean value of the barrier
height, and m inversely proportional to the state variable w according to
m=

12

zI

Lm
,
w

in which L m = ^e 2 ln 2 8rlf o h denotes a constant expressed in J · m, with l standing for the mean value of
the dielectric constant, and f o = 8.85 · 10 -12 F · m -1 symbolising the vacuum permittivity. Finally, in equation
(31) the expressions for B and z I are given by
IEEE cIrcuIts and systEms magazInE

61



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