According to Eq. 3.7, the term for the TiO-based memristor is obtained as

| (6.11) |

From Eq. 3.8, we then obtain

| (6.12) |

Therefore, we get

| (6.13) |

Thus, according to Eq. 6.4 and Eq. 6.5, this device can be used for charge-based memristive sensing. Let us consider a memristor-capacitor (MC) circuit shown in the inset of Fig. 6.5d. Fig. 6.5a shows the characteristics of the memristor, when the MC circuit is excited by a step voltage source where =0 for and 1 for . The memristor exhibits different behavior for different capacitances. As it is expected, the smaller the capacitance the smaller the charging time (Fig. 6.5b) as well as the smaller the charge modulation. Thus, the memristance exhibits smaller modulation for smaller capacitance (Fig. 6.5c). This is the key idea to measure the capacitance by using Eq. 6.4 based on the modulations of the voltage across the capacitor and the memristance. According to Eq. 6.4, if the measurement is performed after the capacitor is fully charged (), only the memristance modulation has to be measured. However, depending on the readout resolution, the measurement can be performed in a time interval (much) shorter than the full charge time (Fig. 6.5d). In Fig. 6.5, the TiO-based memristor device characteristics are assumed as , , , and [69].

Fig. 6.6 shows two magnetoresistive spintronic memristors with current-induced domain wall motion in a giant magnetoresistance spin-valve (Fig. 6.6a) and a tunneling device (Fig. 6.6b). The electrical resistance (conductance) of these devices () is a function of the DW position () in the magnetic free layer [96, 94, 137]. A complete antiparallel alignment results in a high-resistance state (HRS; ) of the device, while a fully parallel alignment places it in a low-resistance state (LRS; ).

For the domain wall giant magnetoresistance (DW-GMR) the memristor is given by [96]

R(r) = | = R_{P}(r) + R_{AP}(r) = rR_{L} + (1 - r)R_{H} | ||

= R_{H} - r(R_{H} - R_{L}) = (1 - rGMR)R_{L}, | (6.14) |

where is the domain wall position, represent the relative DW position, and denotes the length of the free layer. is the giant magnetoresistance ratio. According to Eq. 4.4, the electrical conductance of the domain wall tunnel magnetoresistance (DW-TMR) memristor is obtained as [137]

G(r) = | = G_{P}(r) + G_{AP}(r) = rG_{H} + (1 - r)G_{L} | ||

= (1 + rTMR)G_{L} | (6.15) |

is the tunneling magnetoresistance ratio. The term () is obtained from Eq. 6.16 (Eq. 6.17) for the DW-GMR (DW-TMR) memristors, where is the DW velocity.

| (6.16) |

| (6.17) |

According to Eq. 6.16 (Eq. 6.17), in the DW-GMR (DW-TMR) memristor the term () is a constant and thus is suited for charge- (flux)-based measurement, if there is a linear relationship between the domain wall velocity () and the current (voltage ). It can be shown that a linear dependence of the DW velocity with respect to the applied current or voltage is expected, when the ratio is nonzero, where is the damping parameter and defines the strength of the non-adiabatic spin-torque [174, 175]. In fact, for the DW exhibits a threshold current density required for current-induced motion, even when in the absence of an extrinsic pinning [170, 176]. Thus, only in the presence of non-adiabatic spin-torque (), the DW-GMR and the DW-TMR memristors can be used for charge- and flux-based sensing. In fact, the non-adiabatic spin-torque allows for a non-zero mobility of the DW where the mobility is proportional to the ratio [210, 211]. A comprehensive interpretation of the non-adiabatic spin-torque expressed by is still subject to controversial discussions and there are also difficulties to experimentally characterize [174, 212, 213, 214]. In [213] experimental data shows ratios of and for CoNi multilayers and FePt alloy thin films, respectively.

As an example of a spintronic capacitance sensor, Fig. 6.7a shows the average DW velocity as a function of the current passing through the DW-GMR memristor (connected in series to a capacitor as shown in Fig. 6.1a) based on the one-dimensional model of DW dynamics explained in Section 4.2.2. According to Eq. 6.16, the memristor is suited for capacitance sensing when . Fig. 6.7b illustrates that when the MC circuit (Fig. 6.1a) is excited by a step voltage the final domain wall position in the DW-GMR memristor and thus the memristance value is a function of the capacitance value. The sensitivity increases with the ratio , but at the cost of decreased sensing range. In fact, the increase in the ratio , increases the DW velocity and thus the sensitivity. It should be noted that for , the DW propagation is characterized by a linear regime and a nonlinear regime above the Walker breakdown [215, 174, 216]. Above a critical current value called the Walker breakdown, the DW motion becomes non-uniform (oscillatory) which is unfavorable for memristive sensing as discussed in the following.

The dynamic properties of a propagating magnetic DW are strongly affected by the device geometry [217]. A magnetic thin-film (MTF) element with varying width and constant thickness (Fig. 6.8a) has been proposed as a spintronic (DW-MTF) memristor [61] based upon the STT-induced DW motion in the film-length direction . The reaction force exerted by the domain wall to the electrons acts as the wall resistance [61]. The current-induced DW motion exhibits memristive properties, when the thin-film aspect ratio is varying with . Indeed, when the thickness of the device is fixed, the mobility of the DW and thus the electrical resistance of the device, becomes a function of the thin-film element width and is expressed as [61]

| (6.18) |

where and are the resistance and the width of the element, respectively, when the DW is located at and determines the DW mobility scaling with the aspect ratio as [61]. Therefore, when the spatial dependence of the element width as a function of the DW position is given by

| (6.19) |

the characteristics of the device is obtained as

| (6.20) |

where we have [61]

| (6.21) |

Since the dynamics of the magnetic domain wall motion () is a function of and current density (), the latter equation shows that this device is a memristive system (Eq. 2.14 and Eq. 2.15). For an idealistic case in which the DW velocity is proportional to the current density, the and constitutive relations of the domain wall magnetic thin-film (DW-MTF) memristor are determined as [61]

| (6.22) |

and

| (6.23) |

where and are constant coefficients and determines the spatial dependence of the element’s width on (Eq. 6.19) [61, 172]. According to Eq. 2.10 (Eq. 2.11) and Eq. 6.22 (Eq. 6.23), for charge (flux)-controlled DW-MTF memristor we obtain

| (6.24) |

and

| (6.25) |

Therefore, in order to keep the terms and fixed, the appropriate device geometries for charge- and flux-based memristive sensing methods are (uniquely) determined by

| (6.26) |

and

| (6.27) |

In fact, different device geometries can provide different memristive behaviors required for various sensing applications. With a spatial shape of (), the DW-MTF memristor is suited for charge (flux)-based sensing. Fig. 6.8 shows schematic geometries for memristive sensing where the parameter is supposed equal to [172].

Fig. 6.9 shows an example of inductance sensing by using the DW-MTF memristor with proper geometry based on Eq. 6.9. The applied current is and all axes are dimensionless, with current, memductance, voltage, time, and angular frequency normalized in units of , , , , and . is the memductance when the domain wall position is and is equal to . According to Fig. 6.9a, the memristor shows different memristive behavior for different inductances connected in parallel. Therefore, by using Eq. 6.9 (Fig. 6.9d), the amount of inductance can be calculated when the current (Fig. 6.9b) and memductance (Fig. 6.9c) modulations are known.

Fig. 6.10 and Fig. 6.11 show the simulation results for memristive measurement of time-varying capacitors and inductors. The voltage and current sources are pulse trains with a 50% duty cycle and a magnitude of and and the capacitance and inductance are normalized by and , respectively. () is equal to (), where () corresponds to the memristance (memductance) when domain wall is located at . By sampling and the capacitance (inductance) can be calculated using Eq. 6.4 (Eq. 6.9). Fig. 6.12 shows a DW-MTF memristor structure suited for both capacitance and inductance sensing. The term () is constant for () and the corresponding memristance and memductance are shown in Fig. 6.13.

In order to take a closer look at the DW dynamics in a patterned structure, the one-dimensional model of DW dynamics (Section 4.2.2) is employed to analyze the DW-MTF-based memristive sensing. The dynamic properties of a propagating magnetic DW are strongly affected by the device geometry [217]. The varying width of the DW-MTF element (Fig. 6.12) is taken into account by assuming the current density and the number of spins () in the DW [170] as a function of the DW position . Therefore, Eq. 4.8 is solved for and .

Fig. 6.14 shows the geometry dependence of the DW dynamics for zero and nonzero non-adiabatic spin-torque values. In the absence of the non-adiabatic term (Fig. 6.14a), there is a geometry dependent threshold current required to move a DW. Therefore, for the currents below a current threshold the memristive behavior can not be observed. In the presence of the non-adiabatic term (Fig. 6.14b), the average DW velocity increases linearly with the applied current for low current values. Therefore, with , the memristive behavior is observed and with proper geometry, the device can be used for memristive sensing. Above a (geometry dependent) critical current corresponding to the Walker breakdown [215, 174, 216], the DW motion shows complex behaviors. Therefore, the memristive sensing has to be performed in a low current regime, which for a given sensing resolution, increases the sensing time.