10.1 A Class of Extended Memristors
377
Remark 10.1 It can be easily seen that D ext is not an ideal memristor, nor it can be
brought back to an ideal memristor via a transformation of variables, i.e., D ext is not
an ideal memristor sibling (Chap. 2), when F R is a nonlinear function.
Remark 10.2 It is worth remarking that the assumption F R is Lipschitz near 0
does not imply that there exists the lim v→0 F R (v)/v. As a counterexample, if
F R (v) = G 1 v when v ≥ 0 and F R (v) = G 2 v when v < 0, with G 1 = G 2 ,
then lim v→0+ F R (v)/v = G 1 = lim v→0− F R (v)/v = G 2 .
Remark 10.3 If F R is not locally Lipschitz, then F R (v)/v may be unbounded
near v = 0 (for example, F R (v) = v 1/2 ) and in consequence the zero-crossing
property (10.3) might fail. It is worth to observe that we are interested in using
extended memristors in dynamic nonlinear networks and then it is crucial to assume
that the nonlinear elements are described by Lipschitz functions, otherwise the
uniqueness of the solution for the dynamic equations would be in general not
guaranteed.
Remark 10.4 If the ideal memristor (10.4) and (10.5) and the nonlinear resistor (10.6) are passive, then the extended memristor D ext is passive as well. In fact,
the memductance G(x, v) = G(ϕ M , v) ≥ 0 for all (ϕ M , v) and then the electric
power always enters the memristor.
Remark 10.5 As noticed before, the state variable of D ext is the memristor flux ϕ M .
Several real memristor devices are actually modeled by extended memristors where
nonelectrical physical variables play an important role, e.g., the length of the doped
part of the oxide in MIM structures, the radius of the conductive filament in resistive
memories, or the temperature in phase-change memory devices and thermistors (see
Chap. 2 and also the article [11] for a review).
If the nonelectrical state variables are in a one-to-one correspondence with
the memristor flux ϕ M (see [9, Theorem 2]), as in the case of the linear drift
model (with or without window function) of the HP memristor [12], and of phase
change memories [13], then the memristor device is basically described by an ideal
memristor. As a consequence, such memristor devices fall in the class D ext . On the
other hand, if the state variables cannot be put in correspondence with ϕ M , then the
memristor device may not be modeled via the class D ext . This is true for instance
for the thermistor (Chap. 2), which is a volatile generic memristor, and as such does
not belong to D ext .
Nevertheless, the considered class D ext of extended memristors is of interest and
practical value for many fabricated memristor devices. A relevant case in point is the
real memristors in [14], that have been in fact modeled by the parallel connection of
an ideal memristor and a passive nonlinear resistor (a diode) taking into account the
rectifying effects experimentally observed due to the Schottky barrier at the junction
between platinum and oxide (see also [15]).
Example 10.1 As a specific example, consider for the extended memristor in
Fig. 10.1 the case where the nonlinear resistor is a Shockley diode obeying
377
Remark 10.1 It can be easily seen that D ext is not an ideal memristor, nor it can be
brought back to an ideal memristor via a transformation of variables, i.e., D ext is not
an ideal memristor sibling (Chap. 2), when F R is a nonlinear function.
Remark 10.2 It is worth remarking that the assumption F R is Lipschitz near 0
does not imply that there exists the lim v→0 F R (v)/v. As a counterexample, if
F R (v) = G 1 v when v ≥ 0 and F R (v) = G 2 v when v < 0, with G 1 = G 2 ,
then lim v→0+ F R (v)/v = G 1 = lim v→0− F R (v)/v = G 2 .
Remark 10.3 If F R is not locally Lipschitz, then F R (v)/v may be unbounded
near v = 0 (for example, F R (v) = v 1/2 ) and in consequence the zero-crossing
property (10.3) might fail. It is worth to observe that we are interested in using
extended memristors in dynamic nonlinear networks and then it is crucial to assume
that the nonlinear elements are described by Lipschitz functions, otherwise the
uniqueness of the solution for the dynamic equations would be in general not
guaranteed.
Remark 10.4 If the ideal memristor (10.4) and (10.5) and the nonlinear resistor (10.6) are passive, then the extended memristor D ext is passive as well. In fact,
the memductance G(x, v) = G(ϕ M , v) ≥ 0 for all (ϕ M , v) and then the electric
power always enters the memristor.
Remark 10.5 As noticed before, the state variable of D ext is the memristor flux ϕ M .
Several real memristor devices are actually modeled by extended memristors where
nonelectrical physical variables play an important role, e.g., the length of the doped
part of the oxide in MIM structures, the radius of the conductive filament in resistive
memories, or the temperature in phase-change memory devices and thermistors (see
Chap. 2 and also the article [11] for a review).
If the nonelectrical state variables are in a one-to-one correspondence with
the memristor flux ϕ M (see [9, Theorem 2]), as in the case of the linear drift
model (with or without window function) of the HP memristor [12], and of phase
change memories [13], then the memristor device is basically described by an ideal
memristor. As a consequence, such memristor devices fall in the class D ext . On the
other hand, if the state variables cannot be put in correspondence with ϕ M , then the
memristor device may not be modeled via the class D ext . This is true for instance
for the thermistor (Chap. 2), which is a volatile generic memristor, and as such does
not belong to D ext .
Nevertheless, the considered class D ext of extended memristors is of interest and
practical value for many fabricated memristor devices. A relevant case in point is the
real memristors in [14], that have been in fact modeled by the parallel connection of
an ideal memristor and a passive nonlinear resistor (a diode) taking into account the
rectifying effects experimentally observed due to the Schottky barrier at the junction
between platinum and oxide (see also [15]).
Example 10.1 As a specific example, consider for the extended memristor in
Fig. 10.1 the case where the nonlinear resistor is a Shockley diode obeying