86
2 Fundamental Properties of Mem-Elements
improve the performance of these devices in their diverse applications, clearly
explains why there is still a lot of work to be carried out in memristor modeling,
theory, and simulation. Of course some memristor technologies are more mature
than others, and in some cases the physics underlying the dynamics of resistance
switching memories are well understood, allowing the development of accurate
models, their extensive simulation, and their analysis to establish solid theoretic
foundations on circuits and systems based upon them.
In the following, we summarize various memristor mathematical/circuit models
proposed in the literature. Some models assume that the control waveform is in
current form (the voltage v-current i relationship is expressed by a state-dependent
Ohm’s law), views the memristance as the series between two variable resistances,
associated with the insulating and conductive layers of the nano-film, and sets the
width w of the conductive layer, normalized with respect to the entire length D of
the device, as the state x =
w
D ∈ [0, 1] of the system. An instance of this kind is the
linear drift model from Williams [28] (cf. Sect. 2.4.2.1), where the time derivative
of the state is proportional to the input waveform in current form. Such model is
valid under the assumption of low electric field, since it does not take into account
the boundary behavior.
In the nonlinear drift models from [29, 30] and [59] the rate of change of the
state is proportional to the product between the input waveform in current form
and a window function accounting for nonlinear dynamical behavior and imposing
suitable boundary conditions.
In Joglekar’s model [29] the window function is defined as ζ J (x) = 1−(2x−1) 2p
(p is a positive integer). Such window describes the suppression of dopant drift
close to the extremities, but is not vertically scalable (i.e., its maximum value may
not be up- or down-shifted) and introduces the so-called “terminal-state problem”
[59], since if the state is at either of its two bounds it may not leave it for any
subsequent time instant. Note that for p = 1 Joglekar’s window is a scaled (by a
factor of 4) version of yet another window previously derived by Strukov in [28],
i.e., ζ S (x) = x(1 − x). Benderli [60] presented a circuit realization of Strukov’s
model [28], where the use of comparators and logic gates allowed the emulation of
the state clipping at or release from either bound.
In Biolek’s model [30] the window function depends on both the state x and input
current i, being defined as ζ B (x, i) = 1 − (x − stp(−i)) 2p , where stp(x) = 1 for
x ≥ 0 and stp(x) = 0 otherwise (p is a positive integer). Such window resolves
the “terminal-state problem,” but has limited scalability (in particular, its maximum
value may not exceed +1 [59]). PSpice implementations of Joglekar’s and Biolek’s
models are reported in [30].
In the versatile model proposed by Prodomakis [59] the window function
ζ P (x) = j (1 − ((x − 0.5) 2 − 0.75) p ) has two positive real control parameters
j and p and is vertically scalable, i.e., 0 ≤ max{ζ P (x)} and max{ζ P (x)} 1. A
PSpice version of such model may be easily derived by modifying the PSpice code
file available in [30].
Another model endowed with a PSpice circuit implementation was developed by
Cserey [61]. In this model the state evolution function in Strukov’s model [28] was
2 Fundamental Properties of Mem-Elements
improve the performance of these devices in their diverse applications, clearly
explains why there is still a lot of work to be carried out in memristor modeling,
theory, and simulation. Of course some memristor technologies are more mature
than others, and in some cases the physics underlying the dynamics of resistance
switching memories are well understood, allowing the development of accurate
models, their extensive simulation, and their analysis to establish solid theoretic
foundations on circuits and systems based upon them.
In the following, we summarize various memristor mathematical/circuit models
proposed in the literature. Some models assume that the control waveform is in
current form (the voltage v-current i relationship is expressed by a state-dependent
Ohm’s law), views the memristance as the series between two variable resistances,
associated with the insulating and conductive layers of the nano-film, and sets the
width w of the conductive layer, normalized with respect to the entire length D of
the device, as the state x =
w
D ∈ [0, 1] of the system. An instance of this kind is the
linear drift model from Williams [28] (cf. Sect. 2.4.2.1), where the time derivative
of the state is proportional to the input waveform in current form. Such model is
valid under the assumption of low electric field, since it does not take into account
the boundary behavior.
In the nonlinear drift models from [29, 30] and [59] the rate of change of the
state is proportional to the product between the input waveform in current form
and a window function accounting for nonlinear dynamical behavior and imposing
suitable boundary conditions.
In Joglekar’s model [29] the window function is defined as ζ J (x) = 1−(2x−1) 2p
(p is a positive integer). Such window describes the suppression of dopant drift
close to the extremities, but is not vertically scalable (i.e., its maximum value may
not be up- or down-shifted) and introduces the so-called “terminal-state problem”
[59], since if the state is at either of its two bounds it may not leave it for any
subsequent time instant. Note that for p = 1 Joglekar’s window is a scaled (by a
factor of 4) version of yet another window previously derived by Strukov in [28],
i.e., ζ S (x) = x(1 − x). Benderli [60] presented a circuit realization of Strukov’s
model [28], where the use of comparators and logic gates allowed the emulation of
the state clipping at or release from either bound.
In Biolek’s model [30] the window function depends on both the state x and input
current i, being defined as ζ B (x, i) = 1 − (x − stp(−i)) 2p , where stp(x) = 1 for
x ≥ 0 and stp(x) = 0 otherwise (p is a positive integer). Such window resolves
the “terminal-state problem,” but has limited scalability (in particular, its maximum
value may not exceed +1 [59]). PSpice implementations of Joglekar’s and Biolek’s
models are reported in [30].
In the versatile model proposed by Prodomakis [59] the window function
ζ P (x) = j (1 − ((x − 0.5) 2 − 0.75) p ) has two positive real control parameters
j and p and is vertically scalable, i.e., 0 ≤ max{ζ P (x)} and max{ζ P (x)} 1. A
PSpice version of such model may be easily derived by modifying the PSpice code
file available in [30].
Another model endowed with a PSpice circuit implementation was developed by
Cserey [61]. In this model the state evolution function in Strukov’s model [28] was