54
2 Fundamental Properties of Mem-Elements
Theorem 2.1 (The Nonvolatile Memristor Theorem) A memristive device with a
scalar state variable x is nonvolatile if its POP intersects the x-axis at two or more
points with a negative slope.
Let us now discuss how the property of passivity influences the volatility or
nonvolatility of a memristive device. The following can be shown to hold [6].
Theorem 2.2 (Two Is Infinity) Suppose a passive memristive device is nonvolatile
and let x A < x B be two stable EPs. Then, any ¯
x satisfying x A ≤ ¯
x ≤ x B is also a
stable EP of the memristive device.
The latter theorem can be read as follows: if one can measure experimentally a
passive memristive device that has two different small-signal conductance levels,
then it must also be endowed with a continuum of memory states (i.e., stable EPs).
Note that Theorem 2.2 doesn’t apply to a memristive device with a POP like in
Fig. 2.20a because in that case passivity fails.
2.3.2.1 Examples
A brief collection of examples is included in order to illustrate the concept of volatile
and nonvolatile memory in real memristive devices.
Example 2.15 The Pt/TaO x /Ta memristive device in [16] is a passive voltagecontrolled memristor defined by the state-dependent Ohm’s law
i = G(x, v)v = α(1 − e
−βv ) + γ x sinh(δv)
dx
dt
= g(x, v) =
⎧
⎪ ⎨
⎪ ⎩
λ sinh(ηv)(
1−x
τ 0
), v ≥ 0
λ sinh(ηv)(
x
τ 0
), v < 0
where α, β, γ , δ, λ, μ, τ 0 , η are technological parameters and the scalar state variable is nonnegative and bounded between 0 and 1, i.e., x ∈ [0, 1].
It is worth to note that such memristive device can be seen as a nonlinear resistor
(i.e., α(1−e −βv )) connected in parallel with a memristive device (i.e., γ x sinh(δv));
the whole memductance results to be
G(x, v) =
α
v
(1 − e
−βv ) + x
γ
v
sinh(δv)
.
It can be checked that the memductance is bounded in a neighborhood of (x, 0) for
any x because (see also Example 2.26)
G(x, 0) = lim
v→0
G(x, v) = αβ + xδγ.