64
2 Fundamental Properties of Mem-Elements
The state-dependent Ohm’s law of HP memristor can be written as 8
v(t) =
R on
w(t)
D
+ R off
1 −
w(t)
D
i(t)
where R off is the resistance when w = 0 and the full device is undoped, while R on
is the resistance when w = D, i.e., it is fully doped.
Vacancies (positive ions) can move when subject to a vector field. For simplicity,
a uniform electric field is assumed through the device and it is also supposed a
linear relationship between drift-diffusion velocity and the electric field, that is ions
have a constant average mobility μ V . Such physical assumptions allow to write the
differential equation governing the (scalar) state variable w(t)
dw(t)
dt
= μ V
R on
D
i(t).
(2.35)
By integrating we have
w(t) = μ V
R on
D
q(t)
(2.36)
and, assuming R on R off , it turns out that
v(t) = R(q(t))i(t)
(2.37)
where we have let
R(q(t)) = R off
1 −
μ V R on
D 2 q(t)
.
(2.38)
Equation (2.37), together with
dq(t)
dt
= i(t)
are formally equivalent to (2.29) and (2.30) describing a charge-controlled ideal
memristor.
Waveforms of i(t) and w(t) under a sinusoidal voltage v(t) with angular
frequency ω 0 are shown in Fig. 2.25a. The voltage amplitude is set in such a way
that w(t) belongs to its domain of definition, i.e., w ∈ [0, D]. The corresponding
pinched hysteresis loops for two different angular frequencies ω 0 and 10ω 0 are in
8 The dimensionless state variable x(t) =
w(t)
D is commonly used in several publications. In such a
case x is bounded within the domain [0, 1].
2 Fundamental Properties of Mem-Elements
The state-dependent Ohm’s law of HP memristor can be written as 8
v(t) =
R on
w(t)
D
+ R off
1 −
w(t)
D
i(t)
where R off is the resistance when w = 0 and the full device is undoped, while R on
is the resistance when w = D, i.e., it is fully doped.
Vacancies (positive ions) can move when subject to a vector field. For simplicity,
a uniform electric field is assumed through the device and it is also supposed a
linear relationship between drift-diffusion velocity and the electric field, that is ions
have a constant average mobility μ V . Such physical assumptions allow to write the
differential equation governing the (scalar) state variable w(t)
dw(t)
dt
= μ V
R on
D
i(t).
(2.35)
By integrating we have
w(t) = μ V
R on
D
q(t)
(2.36)
and, assuming R on R off , it turns out that
v(t) = R(q(t))i(t)
(2.37)
where we have let
R(q(t)) = R off
1 −
μ V R on
D 2 q(t)
.
(2.38)
Equation (2.37), together with
dq(t)
dt
= i(t)
are formally equivalent to (2.29) and (2.30) describing a charge-controlled ideal
memristor.
Waveforms of i(t) and w(t) under a sinusoidal voltage v(t) with angular
frequency ω 0 are shown in Fig. 2.25a. The voltage amplitude is set in such a way
that w(t) belongs to its domain of definition, i.e., w ∈ [0, D]. The corresponding
pinched hysteresis loops for two different angular frequencies ω 0 and 10ω 0 are in
8 The dimensionless state variable x(t) =
w(t)
D is commonly used in several publications. In such a
case x is bounded within the domain [0, 1].