1. Introduction

In 1971,Leon Chua postulated the theory of a memristor as the fourth basic circuit element^[1]. Later in 2007,a thin film stack comprising titanium dioxide TiO₂ as the bulk layer material sandwiched between platinum (Pt) electrodes was proposed as a candidate for nonvolatile resistive random-access memory (RRAM) that exhibits resistance switching behavior^[2]. In 2008,a nanoscale resistance switching device was successfully fabricated,and named memristive devices^[3]. These memristive devices exhibited nonvolatile resistance switching abilities that make them a promising alternative to flash memory^[4],dynamic random-access memory (DRAM)^[5] and static random-access memory (SRAM)^[6] applications. Its growing reputation for being a nanoscale nonvolatile device has attracted researchers into developing RRAM applications^{[5, 6]},improved logic gates^[7],designing memristive neuromorphic systems^[8],and improving memory density^[9].

Although preliminary experimental results show that memristors can replace memory devices in the future,the endurance of memristors is still less than that of a typical DRAM (10¹² cycles)^[10]. The highest number of endurance cycles achieved by a TiO₂ memristive device is 2×10⁶ cycles using a TiN electrode^[11],and 10⁴ cycles using Pt electrodes^[12],where both are far from the endurance benchmark of current established memory devices^[13].

Recently,researchers have shown that other metal oxides could also be used as the bulk material to fabricate nanoscale memristive devices,such as zinc tin oxide^[14],vanadium oxide^[15],silicon oxide^[16],copper oxide^[17],hafnium oxide^[18],and tantalum oxide (TaO₂) ^[19]. However,TaO₂ is considered as one of the prospective resistive switching materials due to its two stable phases (TaO₂ and Ta₂O₅)^[20],which can control the stability of its high and low resistance states. In addition,the TaO₂ physical memristive devices have exhibited better resistance switching characteristics than TiO₂ memristive devices,in terms of switching speed (5 times faster) and better endurance (10⁸ times more switching cycles)^[21],longer retention (10 years) and larger resistance ratio. Due to these statistics,a comparison between TaO₂ and TiO₂ physical memristive devices was conducted to understand the requirements and factors to fabricate better performing memristive devices. TiO₂ memristive devices were selected as the benchmark because the study and fabrication of TiO₂ memristive devices have been extensive.

Several review articles have provided a general overview of the memristive devices that are found in References ^[22-24]. This research work differs from other memristor review articles because this article provides a direct comparison between two types of memristive devices that differ only in the substrate. The comparisons performed in this article are in the aspect of physical structures,electroformation,switching mechanism,switching speed,switching energy,resistance retention,and device endurance. The review in this article investigates the physical and chemical properties of TiO₂ and TaO₂-based memristors,and the changes that occur in both memristive devices during electroformation and switching. The properties of the device structures that could lead to possible future discovery of new bulk layer materials or improved structures were also studied. This article provides a review on the comparison of the performances of TiO₂ and TaO₂ physical memristive devices in the literature,and discusses the methods which could improve retention,endurance and switching performance of memristive devices. The material and geometric factors that affect switching speed and resistance ratios are also explained. This research work provides several recommendations,summarized from previous work to fabricate better performing memristive devices. This article concludes the physical and material factors that improve the electrical performance of memristive devices.

This article is organized as follows: Section 2 describes the electroformation process in general and also specifically for both structure types of memristive devices; Section 3 explains the switching mechanisms,and compares the switching speeds and energy requirements; Section 4 discusses resistance state non-linearity,resistance state stability and resistance ratio; Section 5 explains the difference in retention capability and endurance of the memristive devices; Section 6 studies the effect of varying electrode cross-sectional area and bulk thickness; and Section 7 summarizes this article. This article is also an extension of the article in Reference ^[25].

2. Structure of TiO₂ and TaO₂ memristive devices

TiO₂ and TaO₂ memristive devices have different physical structures (Figure 1). The TiO₂ memristive device has a bulk layer of TiO_2-X sandwiched between two Pt electrodes,resembling metal-insulator-metal (MIM) structures with the Pt electrodes being metal and TiO₂ as an insulator^[2]. For TaO₂ memristive devices,the TaO_2-X bulk layer is formed by reactive sputtering of a Tantalum (Ta) layer in oxygen,and a Ta₂O_5-X layer is fabricated by using oxygen plasma with Ta metal above the TaO_2-X bulk layer^[26]. The oxygen plasma conditions ensure that the Ta metal is oxygen rich to form the Ta₂O_5-X layer. The top Pt electrode is directly deposited over the Ta₂O_5-Xlayer^[27],giving the substrate of the TaO₂ memristive devices two layers instead. The purpose of the insulating layer of Ta₂O₅ is to convert the device into a bipolar resistive switching device^{[28, 29]} and also results in TaO₂ memristive devices resembling metal-insulator-semiconductor-metal (MISM) structures.

However,by using the Ta₂O₅/TaO₂ RRAM structure instead of MIM structure and by selecting each of these layers to have an appropriate resistance,two important factors can be controlled: the programming current and device stability^[26]. The MISM memristor structure reduces the low resistance state current by maintaining a medium resistance during switching between MISM and metal-metal-semiconductor-metal (MMSM),which is the low resistance state of the MISM structure^[26]. The low-current operation devices (<100μA) are necessary for more effective RRAM applications^[20]. In addition,the Ta₂O₅ MIM structure is not suitable for bipolar resistive switching due to the highly resistive Ta₂O₅ forming a very high Schottky barrier with the metallic electrodes. Thus,a reverse bias will not be able to overcome the high Schottky barrier. Therefore,in an MISM structure,the interface between the metallic electrode and semiconductor TaO₂ bulk layer is reported to form another Schottky barrier that will decrease the high Schottky barrier,which assists resistance switching,and dominates the relationship between current and voltage for MISM memristor structures^[30].

In the TaO₂ memristive devices,the TaO₂ layer functions as an oxygen vacancies supply layer whereas Ta₂O₅ is an oxygen vacancies accumulation layer^[31]. Although the fabrication of TaO₂ memristive devices requires an additional layer of Ta₂O₅,but it serves as an advantage against other MIM structured memristive devices because the variation of the thickness of the Ta₂O₅ insulator layer provides an easier manipulation of the resistance ratio of TaO₂ memristive devices^[26].

Despite having different device structures and bulk layer materials,both TiO₂ (MIM) and TaO₂ (MISM) memristive devices exhibit similar resistive switching behavior. This is evident by the fact that both devices are able to switch between high and low resistance states and the switching of states depends on the direction of current (demonstrating bipolar switching behavior). However,the switching mechanisms are different due to the different device structures. In short,the difference in electrical performance is mainly due to the physical structure and chemical properties of bulk layer material of TiO₂ and TaO₂ memristive devices.

^*Please note: hereafter,TiO₂ and TaO₂ memristive devices will respectively be written as TiO₂ and TaO₂ devices.

3. Electroformation

3.1 Electroformation

Post-fabricated MIM structures cannot conduct electricity because insulator bulk layers have high resistance against electrical conduction under normal operating conditions. To form a device capable of conducting current,electroformation is performed,which is an irreversible one-time process that forms a device capable of conducting current,usually by creating conducting channels through a high resistance layer. This is done by causing physical changes to the device using electrical potential. It is essential for MIM structures to undergo electroformation in order to exhibit memristive behavior^{[32, 33]}.

The magnitude of electroforming voltage is usually much larger than the switching voltage magnitude required for subsequent resistance switching to take place^[34]. The electroforming voltage magnitude also usually exceeds the breakdown potential of the device,causing a build-up or depletion of oxygen ions at the electrodes,depending on the polarity of electroforming voltage. The subsequent physical switching mechanism takes place around these erupted regions because it was observed that conducting channels are formed and collapse near these regions^[35].

During electroformation,the potential bias applied at the electrodes energizes the oxygen ions of negative charge to drift to one of the electrodes of the device. Oxygen ions will combine to form oxygen gas molecules. The oxygen gas molecules are trapped in the interface between the electrode and bulk layer. Gas pressure builds up as more oxygen gas

molecules are formed. This eventually causes an eruption through the top electrode and a permanent physical deformation occurs in the memristive device^[12].

Although there is permanent physical change to the device during electroformation,the subsequent resistance switching after electroformation is not affected. This is due to the subsequent conducting channels that are responsible for the conduction of current being formed and collapsed near the electroformed conducting channel^{[36, 37]}. Thus,no further deformations at other regions are required^[37].

3.2 Gibbs free energy of formation

There are various mechanisms proposed to explain the theory of electroformation in MIM devices. According to Reference ^[34],the electroforming voltage is given by:

${{V}_{F}}\cong \Delta {{G}_{MX}}xF$

(1)

where ΔG_MX is the Gibbs free energy of formation of the bulk layer material and F is Faraday's constant. The proposed theory by Greene et al. states that the electroforming voltage is independent of bulk layer thickness^[34]. This theory can be used to compare two devices of similar physical dimensions but different bulk layer material. According to Equation (1),the electroforming voltage required is proportional to the Gibbs free energy of the formation of the bulk layer material. The Gibbs free energy of the formation of metal oxides commonly used in the bulk layer of MIM resistive switching devices is shown in Table 1^[38]. From Table 1,it was noticed that the Gibbs free energy of the formation of TaO₂ and Cu₂O is among the lowest whereas HfO and TiO₂ is among the highest. TaO₂ is more widely used over Cu₂O due to its more stable resistance states as seen in the current-cycle and resistance-cycle graphs of TaO₂ and Cu₂O memristive devices^{[17, 30, 39]}. Other researchers have also shown that lower differences of the Gibbs free energy of the formation between TaO₂ and Ta₂O₅ show more stable behavior^[20].

Although memristive devices with bulk layers of high Gibbs free energy of formation may also exhibit resistive switching behavior,these devices require an additional one-time electroformation step. This is reflected by TiO₂ devices requiring an electroformation process that requires a large electroforming potential before TiO₂ devices can exhibit resistive switching behavior. As TiO₂ devices mimic an MIM structure,many publications have reported that other MIM structures of different materials also requires an electroforming step before it exhibits memristive behavior^{[40, 41]}. Conversely,TaO₂ devices with similar physical dimensions as TiO₂ devices would require ∼ 4.5 times less energy for the electroforming process because the Gibbs free energy of the formation of TaO₂ is ∼4.5 times smaller than the Gibbs free energy of the formation of TiO₂. Moreover,if the Gibbs free energy of the formation is very small,the electroforming voltage may become smaller than the switching voltage of the memristive device^[42]. The damage to the device is insignificant and the electroformation process could be part of the first switching process of the device. This avoids the need of a large voltage for a separate electroforming process.

Research also shows that the electroforming voltages of physical TiO₂ and TaO₂ devices are proportional to bulk layer thickness^{[28, 43]}. This may be in contradiction to Greene's theory,but it is noted that if devices are of the same physical dimensions,then Greene's theory is completely applicable. If two devices have different physical dimensions but similar bulk layer material,then using the Dearnaley theory would be more appropriate.

3.3 Electroformation according to Dearnaley et al

Dearnaley et al. proposed that the electroforming voltage is proportional to the insulator thickness in MIM devices^[44]. They reasoned that this is due to the insulator-metal surface not being completely smooth. Thus,there will be certain regions where the bulk layer is thinner than other regions. Voltage applied at the electrodes is applied across the entire bulk layer surface. The electric field applied during electroforming,E^F is the amount of voltage supplied per insulator thickness,given by:

${{E}_{F}}=\frac{V}{L}$

(2)

where V is the voltage applied across the device and L is the bulk material thickness of the device. Electric field is inversely proportional to insulator thickness,thus the electric field is larger at those thinner bulk layer regions. At these high electric field regions,electroformation is more likely to occur and hence forms a conducting channel at these regions. This theory also explains why conducting channels in TiO₂ and TaO₂ devices are not spanned across the entire electrode area^[45] This is discussed further in section 4 of this article.

The electroforming field applied across the bulk layer is inversely proportional to the bulk layer thickness. Hence,insulators thicker than 1 $\mu$m cannot be electroformed due to insufficient electroforming field across the bulk layer^{[2, 34]},whereas devices thinner than 10 nm may suffer from permanent dielectric breakdown due to the high electric field^[46],or due to large amounts of current flow^{[47, 48]}.Therefore,to avoid a permanent breakdown or excess physical damage during electroformation,current compliance methods are commonly used^[49].

3.4 Avoiding electroformation in memristive devices

In order to avoid electroformation in TiO₂ devices,two methods were proposed: (i) by thinning the bulk layer to a few nanometers thick^[12],and (ii) depositing the bulk layer by reactive sputtering in an oxygen rich ambient^[50]. Reducing the device size also decreases the physical damage caused by oxygen gas eruption because smaller devices increase the rate of the release of oxygen molecules out of the device^[12].

To avoid electroformation in TaO₂ devices,the top electrode Pt is proposed to be replaced with Ta^[21]. Although a different electrode is used,the device still behaves with similar memristive properties^[42]. Using the magnetron sputtering method,TaO₂ devices are able to exhibit memristive behavior without electroformation^[31]. This shows that TaO₂ devices can exhibit memristive behavior without the need for an additional electroformation process. It also reduces energy consumption by avoiding the need of a large voltage application to electroform the device.

4. Switching mechanism

During the fabrication process,oxygen ions are injected into the memristive devices in order to dope the bulk layer. This process is uneven and the oxygen ions are asymmetrical in the bulk layer^{[51, 52]}. It is the asymmetric distribution of oxygen ions (or vacancies) in the TiO₂ bulk layer that contributes to the memristive properties of the device. The defect states induced by oxygen vacancies act as the n-type dopant in TiO₂ and the highly resistive TiO₂ layer becomes more conductive. The asymmetric distribution of oxygen vacancies also causes the TiO₂ layer to have regions of high and low resistances. The highly-doped region is less resistive,while the lowly-doped region is more resistive. Due to the asymmetric doping of the bulk layer,only one of the electrodes dominates the switching mechanism,which is the electrode that is nearer to the lowly-doped region of the bulk layer. The formation and collapsing of conducting channels takes place nearer to this electrode^[53]. The region where switching cycles will take place,or the electrode that dominates the switching cycles,is determined during fabrication^{[26, 54]}.

4.1 TiO₂ device switching mechanism

The switching mechanism in TiO₂ devices involves the dislocation of the mobile charged species ions or vacancies towards or away from an electrode. This process either forms or collapses conducting channels^[55]. Drifting of oxygen vacancies from the region of highly-doped TiO₂ towards the oxygen-deficient region will form areas of lower resistances in the lowly-doped region of the bulk layer^{[56, 57]}. These areas of lower resistances will then combine to form a chain of low resistance regions,which creates the conducting channels. This resembles the switching mechanism of other MIM memristive devices^[58].

Positive charge oxygen vacancies are migrated in the bulk layer during switching (Figure 2(a)) forming a filament-like structure,protruding the entire bulk layer (Figure 2(b))^[59]. The formation of the filament-like structure conducting channel is localized and does not take place throughout the entire cross-section of the bulk layer^[45]. This filament-like structure in the bulk layer is the conducting channel consisting of the TI₄O₇ phase^[41]. It is less resistive than the non-doped regions of the bulk layer,which are composed of the TiO₂ phase^[55]. The opposing movement of oxygen vacancies transforms the TI₄O₇ phase in the conducting channel into other Ti-O phases,mainly the TiO₂ phase (Figure 2(c)). The high resistance TiO₂ phase does not conduct current and the conducting channel becomes highly-resistive. The collapse of the conducting channel eliminates the current conduction path from the bulk layer,so large amounts of current cannot tunnel through the device (Figure 2(d)).

4.2 TaO₂ device switching mechanism

In TaO₂ devices,the resistive switching mechanism is assisted by a very thin insulating layer of a highly resistive Ta₂O₅ phase fabricated between the top electrode and bulk layer^{[21, 28, 30]}. The insulating layer of Ta₂O₅ consists of both Ta₂O₅ with a very small percentage of the TaO₂ phase.

To turn the device ON,a negative bias is applied at the top electrode and the bottom electrode is grounded. The negative bias at the top electrode repels negative charged oxygen ions away from the top electrode (Figure 3(a)). Ta₂O₅ molecules are reduced to TaO^[60] and Ta^[61] molecules by releasing oxygen ions. The oxygen ions drift towards the TaO₂ bulk layer. After a number of TaO and Ta molecules align in the Ta₂O₅ layer to form a conducting channel that is TaO and Ta-rich,large amounts of current can tunnel through the insulating layer. This switches the device into LRS (Figure 3(b)).

To turn the TaO₂ device off,a positive bias is applied at the top electrode while the bottom electrode remains grounded. This causes the attraction of oxygen ions from the TaO₂ bulk layer towards the top electrode (Figure 3(c)). As the oxygen ions accumulate in the insulating layer,oxidation takes place and TaO and Ta molecules are oxidized into Ta₂O₅ molecules. The number of Ta₂O₅ molecules in the conducting channel increases and the resistance of the conducting channel increases. The conducting channel collapses when a number of Ta₂O₅ molecules disconnect the conducting channel. This prevents current from flowing through the bulk layer and Ta₂O₅ layer,thus the device is in the HRS (Figure 3(d)).

4.3 Comparison of switching mechanisms

The switching mechanisms of TiO₂ and Ta₂O₅ devices represent the general switching mechanisms of MIM and MISM devices respectively. It was shown that both MIM and MISM devices use the redox reactions of metal oxides. Metal oxide phases have varying resistivity characteristics. Thus,by performing oxidation and reduction in memristive devices,the conductivity of the bulk layer can be altered.

The major difference between the two types of switching mechanisms is the manner of the formation of the conducting channels. In MIM devices,the conducting channel spans the entire length of the bulk layer of the device,protruding from one electrode to the other. As for MISM devices,the conducting channels are formed in the higher resistance region of the bulk layer^[62]. The lower resistance of the bulk layer does not require the formation of a conducting channel in order to be able to conduct current.

Apart from the mechanism formation and collapsing of conducting channels in the substrate of memristive devices,the location of the formation of conducting channels is also different in MIM and MISM devices. In MIM devices,switching takes place at the interface of metal-insulator that is less doped with oxygen vacancies^{[53, 63]}. Researches have also shown that insertion of nano-interface layers into MIM devices improves resistance switching by acting as an oxygen sink^[64-66],as well as to reduce switching currents^[60]. As for MISM devices,there are also theories that claim the metal-insulator (or metal-semiconductor) interface is responsible for memristive behavior. This metal-insulator (or metal-semiconductor) interface forms a Schottky barrier that is crucial for resistance switching in MISM devices^{[26, 57, 67]}. This type of switching mechanism is further described in References ^{[68, 69]}.

4.4 Switching speed of TiO₂ and TaO₂ memristive devices

Generally,switching causes the movement of charged species (e.g. oxygen ions or vacancies) along the length of the device in both switching mechanisms. A conducting channel is formed as soon as there are sufficient oxygen vacancies at a localized switching region. Conversely,if the number of oxygen ions to metal ions (oxygen vacancies in our case) becomes larger than some critical value,Mott-type metal to insulator transition occurs in the switching layer and the conducting channels are collapsed. Due to the charged species having finite mobility,the application of potential difference at the electrodes must exceed a magnitude (known as the switching voltage) for the mobile charged species to have sufficient energy to overcome potential barriers. The potential difference must also be applied longer than a certain amount of time for sufficient charged species to be displaced. Hence,the switching speed of memristive devices is defined as the rate of the device to completely switch resistance states. Switching time is measured starting from when voltage is applied across the device electrodes until switching is complete.

The mobility of charged species through the switching layer is dependent on the material type and the forming field across that layer. The mobility of the mobile charged species in both TiO₂ and TaO₂ devices is compared between the devices. The oxygen vacancies mobility μ in TiO₂ is 10¹⁰ cm²/(V⋅s)^[3] and in TaO₂ is 5.4610¹⁰ cm²/(V⋅ s)^[70]. Due to the vacancy mobility in TaO₂ being ∼5 times faster than that in TiO₂,it is deduced that the migration of mobile charged species in TaO₂ is ∼ 5 times faster than that in TiO₂ for similar device dimensions and voltage applied. This means that the time taken to form a conducting channel in TaO₂ devices is ∼ 5 times faster than that for TiO₂ devices,provided the same amount of voltage is applied and devices are of a similar size. Thus,it is concluded that the average switching speed of TaO₂ devices is ∼ 5 times faster than that of TiO₂ devices.

4.5 Switching energy of TiO₂ and TaO₂ memristive devices

The amount of electric field applied and charge carrier mobility are considered when estimating the energy consumed during the switching cycles in the devices. The drift velocity of charge carriers,$v_{\rm d}$ is given as:

${{v}_{d}}=\mu {{E}_{L}}$

(3)

In order to maintain similar switching speeds (fixed drift velocity of charge carriers) the required application of electric field across the length of the bulk layer is inversely proportional to the charge carrier mobility in the bulk material. Assuming TiO₂ and TaO₂ devices of similar physical dimensions,it is postulated that to obtain similar switching speeds,the energy requirement of TiO₂ devices is larger than that of TaO₂ devices. This is due to the charge carrier mobility in TiO₂ being smaller than that in TaO₂.

5. Variable resistance

Since the electrodes and substrate of TiO₂ and TaO₂ devices are in series,the instantaneous resistance of \,the devices is the summation of the resistance of each layer. The resistance of the electrodes is always constant because it does not take place in the switching mechanism. The device substrate has different resistance behavior due to the different switching mechanisms in MIM and MISM devices. Although the resistance changes differently depending on device structure,the physical condition of the bulk layer still depends on the history of charge flow through the devices,i.e. HRS or LRS. Thus,the instantaneous resistance of a memristive device depends on both the history of charge flow and its switching mechanism.

Among the results from fabricated memristive devices,the current-voltage curve of each device varies. It is also noticed that the fabricated memristive devices have varying fabrication processes and device dimensions. This tells us that it may be difficult to apply one current-voltage equation for all memristive devices. The current-voltage equation would also have to include other factors such as device size and material properties. Various theories were proposed to simulate the current-voltage characteristics of the devices and in this article,the types of theories that were proposed hitherto and the justifications for those theories are briefly explained.

5.1 Variable resistance in MIM devices

Drifting of oxygen vacancies along the bulk layer of MIM devices causes the resistance of the bulk layer to change. Initial models assumed that the bulk layer is divided into regions of high and low resistances. The length of the low resistance region is termed as the barrier width,w and is modeled to sweep along the bulk length,$L^[55]. The drifting of oxygen vacancies or barrier width along the bulk length was proposed to be linear^[3]. The rate of change of barrier width is defined as:

$\frac{dw\left( t \right)}{dt}=\frac{\mu {{R}_{ON}}}{L}i\left( t \right)f\left( t \right)$

(4)

where i(t) is the amount of current passing through the device and f(t) is a window function with respect to time. The purpose of the window function is to model the behavior of the barrier width when approaching electrode boundaries. A state equation was proposed by Strukov^[3] to relate barrier width w(t) and instantaneous resistance of the device:

$R\left( t \right)={{R}_{ON}}x\left( t \right)+{{R}_{OFF}}\left[ 1-x\left( t \right) \right].$

(5)

where R_ON is the resistance of the device when it is in LRS and R_OFF for when the device is in HRS. x represents the ratio of the thickness of the low resistance region in the bulk layer (w) to the total thickness of the bulk layer (L),giving x=w/L,where x ranges from 0 to 1. The assumption for this equation is that the bulk layer only contains TiO₂ and TI₄O₇ phases. The authors of Reference ^[3] also assumed that the device could be in a state of intermediate resistance,with conductivity between LRS and HRS.

However,non-linear simulation models have exhibited current-voltage curves which are closer to the physical devices than linear models. The drifting of oxygen vacancies is non-linear in physical devices because the electric field across the device is not linear along the bulk layer due to the asymmetrical distribution of oxygen vacancies in the bulk layer^[71]. Furthermore,the drift velocity of charge carriers is strongly affected by the non-linear resistance along the bulk layer. At the electrode boundaries,the charge carriers move into another medium of different charge and material density. This gives rise to a change in charge mobility and causes the drift velocity of the charge carriers to change as they approach or leave electrodes,which adds to the factor of non-linearity^[72]

Window functions f(x) were subsequently proposed to model the mobility of charge carriers approaching electrode-bulk layer boundaries. The window functions proposed are summarized in Table 2,in chronological order. The window function to simulate the linear drift model was proposed in 2008^[3]. The window function was improved by introducing a parameter p for generalization of non-linear behavior^[71]. However,both of these two window functions will cause the simulation of barrier width to be stuck at 0 or 1,and cannot be adjusted further. To avoid this,a window function was proposed to be dependent on current^[73],where

$stp\left( i \right)=\left\{ \begin{matrix} 1, & i\ge 0 \\ 0, & i<0 \\ \end{matrix} \right.$

(6)

Although the window function of Reference ^[73] has solved the problem issues and has parameter p for curve-fittings,it lacks scalability. Prodromakis $et\,al$. proposed a new window function that improves scalability^[74] giving a range of f_max where $0\leqslant f_{\mathrm{max}}(x)\leqslant 1$ .

Up to this time,there are no physical devices that correlate to any of the proposed windows functions. Conversely,Yao et ,al.^[16] and Rossel et al.^[75] suggested close relations between physical memristive devices and the Simmons tunneling current density equation. Due to the window functions hitherto not fitting the Simmons tunneling current density equation,a window function which correlates to the Simmons equation was proposed^[76]. The Kvatinsky window function fits the Simmons equation but it lacks scalability and does not guarantee symmetrical behavior. This means that the rate of change of drift velocity of charge carriers moving towards one electrode may be different from moving towards the opposite electrode. Two more fitting parameters were also introduced in the form of a_on and a_off,which are used to fit the model to match the Simmons equation. The Simmons tunneling current density equation is given as:

$J=\int_{0}^{{{E}_{m}}}{D\left( {{E}_{x}} \right)}\xi d{{E}_{x}},$

(7)

where $\zeta$ is defined as:

$\xi =\frac{4\pi q{{m}^{2}}}{{{h}^{3}}}\int_{0}^{\infty }{\left[ f\left( E \right)-f\left( E+eV \right) \right]}d{{E}_{r}},$

(8)

and D_Ex is a function of the probability of an electron being able to overcome the potential barrier posed by the insulator layer and penetrate through the substrate from one electrode to another. f(E) and f(E+eV) are the Fermi-Dirac functions for electrons following and going against the potential barrier respectively,while dE_r is the integral with respect to the effective electron direction. The Simmons tunnel equation is the most accurate due to its consideration of distribution and drifting of electrons in both directions. However,the probability function D_Ex is ambiguous due to a lack of experiments performed in obtaining it and may be difficult to obtain^{[12, 77]}. Based on these findings,more study on MIM devices is required in order to produce a generic current-voltage equation for the behavior of TiO₂ devices.

5.2 Variable resistance in MISM devices

$J\mathrm{=}A^{\mathrm{\ast }}T^{\mathrm{2}}\exp\left( \mathrm{-}\alpha _{\rm T}d\sqrt {q\phi_{\rm T}} \right)\exp\left( \frac{\mathrm{-}q\phi_{\rm B}}{kT} \right)\left(\exp\frac{qV}{\eta kT}\mathrm{-1} \right).$

(9)

Equation (9) involves the thermionic-emission for the Schottky barrier and includes a tunneling probability term, ${\rm exp}\left( -\alpha_{\rm T}d\sqrt {q\phi _{\rm T}} \right)$ . A^* is the Richardson constant,q is electron charge,V is voltage applied,T is temperature,k is the Boltzmann constant, $\phi_{\rm T}$ is the effective barrier height,d is insulator layer thickness,and $\alpha _{\rm T}$ is the base transport factor term. $\alpha_{\rm T}$ approaches 1 if the effective mass in the insulator is equivalent to the free electron mass. The Schottky barrier height $\phi_{\rm B}$ term exists in the tunneling current density equation due to the movement of charge carriers between metal and semiconductor through a very thin insulator layer mimicking that of a Schottky-barrier diode. Ideality factor, $\eta$ is close to the value of 1 when the amount of doping in the insulator layer is small,and becomes a larger value when doping increases. Using this equation,simulation models have been developed and they match the characteristics displayed by physical TaO₂ devices^{[68, 79]}.

The switching resistance ratios (R_OFF/R_ON) of TiO₂ devices range from 500 to 10⁵,^{[50, 51, 53, 80, 81]},while the switching resistance ratios of TaO₂ devices range from 10 to 10⁶,^{[19, 30, 42]}. It is estimated that the average resistance ratio of future fabricated TaO₂ devices will be higher than that of TiO₂ devices due to the resistance ratios of TaO₂ devices reaching 10⁶,while TiO₂ devices reach up to only 10⁵ of resistance ratios. Comparing band gaps of TiO₂ and TaO₂/Ta₂O₅,the literature reported that wider band gaps are advantageous for resistive switching^[82]. The band gap of TaO₂ is 4.0-4.2 eV^[83],Ta₂O₅ is 3.9-5.3 eV^{[84, 85]},and TiO₂ is 3.0-3.2 eV^{[82, 86]}. Therefore,this adds weight to the choice of TaO₂ devices over TiO₂ devices.

In order for memristive devices to be able to replace physical memory devices,they need to be able to consistently switch between resistance states. Although a slight deviation of the resistance value is allowed,a stable R_OFF/R_ON ratio is required. In both devices,the HRS will vary much more than the LRS,with an example of resistance switching given in Figure 4. The HRS is formed by collapsing conducting channels,which is not a consistent process because the amount of charge carriers needed to collapse a conducting channel may vary during each switching. In the LRS,the conducting channel effectively shorts the device,so the only resistance along the device is the resistance of the electrodes,which is constant^{[87, 88]}. Most of the TiO₂ and TaO₂ devices have exhibited an almost consistent R_OFF/R_ON resistance ratio^{[30, 89]}. Therefore,both TiO₂ and TaO₂ devices are two of the suitable candidates of memristive devices to replace physical memory devices in terms of stable resistive states.

6. Retention and endurance

Retention and endurance are common measures of the reliability of memory and memristive devices. Retention is the ability of memory and memristive devices to retain logic information,measured by the amount of time that the memory is still retrievable. Endurance is the measurement of the number of times a device can perform a successful read and write process. It is equivalent to the number of switching cycles that the device can perform successfully.

6.1 Retention time of memristive devices

The retention time of a memristive device is defined as the time taken for the device to deteriorate from its resistance state (HRS or LRS) by a certain amount of percentage,R%. The threshold for the percentage of change,R% is determined by researchers or manufacturers. Smaller values of R% yield a better quality of retention in exchange for a shorter retention period. Conversely,larger values of R% result in a longer retention period but yield a device with a poor retention quality.

The retention time of memory devices at room temperature or normal operating conditions is predicted using extrapolation of the Arrhenius plot. This method has been used by many other researchers^[90-92]. The Arrhenius equation defines the rate constant $\tau$ of a chemical reaction as:

$\tau \mathrm{=}A\me^{\mathrm{-}E_{\rm a}\mathrm{/}kT},$

(10)

where E_a is the activation energy of the reaction. A is a pre-exponential factor or frequency factor that depends on the frequency of collision between molecules,while the rate constant τ is the frequency of a successful reaction between reactants. Taking the natural logarithm,Equation (10) becomes:

$\ln \tau {=-}\frac{E_{\rm A}}{k}\left( \frac{{1}}{T} \right){+}\ln A.$

(11)

This method uses temperature-accelerated degradation of memristive devices and the percentage of change in resistance or conductivity,ΔR is observed. The device is degraded past R% at a specific constant temperature T. A graph of ΔR is plotted against time,and the gradient is the rate constant τ for temperature T (Figure 5). A similar procedure is repeated to obtain rate constants at different temperatures.

Temperature-accelerated degradations are usually performed at temperatures much higher than room temperature or normal operating conditions,in the range between 150 ℃ to 250 ℃^[90-92]. The Arrhenius plot is then drawn,where $\ln\tau$ is plotted against $T^{-1}$ (Figure 6). The graph is then extrapolated until the rate constant at room temperature or at normal operating conditions is found.

This approach was used to obtain the retention time for TaO₂ devices operating between 25 ° and 85 °,which is of over 10 years^[30]. This retention time meets the standards of manufacturers of memory devices^[90]. At the time of writing this article,the retention of TiO₂ devices has not yet been obtained using extrapolation of the Arrhenius plot. There were reports stating that TiO₂ devices have a retention time as long as 10 years but there are no experimental data or methods that could provide sufficient evidence to support this statement. Although different fabrication methods and/or slightly different electrodes were tried,experiment data could still not provide sufficient evidence for retention of 10 years^{[11, 93-95]}. Due to the lack of robust methods in obtaining the retention period of TiO₂ devices,it cannot be concluded yet that TiO₂ devices have the retention ability of 10 years.

6.2 Endurance of memristive devices

The endurance of memristive devices is measured by the number of resistance switching cycles that can be successfully performed until the resistance ratio converges towards unity,or when the resistance ratio becomes less than a required amount. The minimum resistance ratio is determined by the application that uses the memristive device. The method to test device endurance is by switching the device continuously until it fails. Recent publications have shown that TaO₂ devices (10¹⁰ cycles^[31]) have better endurance than TiO₂ devices (10⁴ cycles^[96]). It was also shown that after 10⁴ cycles,the resistance states of TiO₂ converge and no longer exhibit memristive properties^[21],while endurance of TaO₂ devices can reach up to 10¹² cycles via increasing the device size^[26]. Increasing the device size improves endurance due to the larger space available to produce more conducting channels. In larger devices,if for any reason a conducting channel fails to reconnect while turning-ON,then the turning-ON voltage will electroform another new conducting channel in the bulk layer. Thus,ensuring another switching cycle can occur.

TaO₂ devices exhibit better endurance than TiO₂ devices due to the multiple stable Magnéli phases of titanium oxide (Ti-O)^[97] compared to the tantalum oxide (Ta-O) series that only has two stable Magnéli phases^[98]. Figures 7 and 8 show the phase diagrams of Ti-O and Ta-O. Magnéli phases of oxides appear when crystallographic shear occurs that changes the stoichiometry of the cation (metal) but still maintaining the coordination requirements of the cation. The homologous series of metal-oxide phases are made up of Magnéli phases of metal-oxides^[99]. Due to the unstable TI₄O₇ phase and multiple Magnéli phases in the Ti-O series,the doping of TiO₂ results in the formation of oxide phases other than TI₄O₇^[41]. After a number of switching cycles,the redox reaction that takes place in the bulk layer forms Ti-O phases other than TiO₂ and TI₄O₇. The other phases of Ti-O are also in thermal equilibrium with TiO₂ and TI₄O₇,which increases the reluctance of the other phases to react. With the increase of other higher resistive phases in the bulk layer,the device will not be able to switch back to its initial R_ON and R_OFF states. This means that the TiO₂ device has reached its endurance limit.

For TaO₂ devices,the Magnéli series consist of only two stable phases: TaO₂ and Ta₂O₅. This means that every chemical reaction in the bulk layer will only involve either of these two Ta-O phases,which improves device endurance. Therefore,it is concluded that TaO₂ devices have better endurance than TiO₂ devices due to the lesser number of stable Magnéli phases in the Ta-O series.

Apart from using materials with a lesser number of stable Magnéli phases,the endurance of resistive switching devices can be improved by regulating the electroformation process^[29]. By minimizing the electroforming process,physical damages are minimized. Thus,preserving the original structure of the device and improving the endurance of the device in the physical aspect,as previously discussed in section 3 of this article.

7. Geometric effect on electrical properties

The thick bulk layer in MIM memristive devices prevents conducting channels from being formed due to the large amount of electric field required to breakdown the thick bulk layer. This is shown by the decrease in probability for the formation of a conducting channel with an increase in bulk layer thickness^[89]. The range of bulk layer that consistently exhibits memristive behavior is below 500 nm^[82],while bulk layers thinner than 50 nm[41,\thinspace 100] or even 10 nm^[101] can also show memristive properties. Switching voltages are observed to be independent with varying bulk layer thickness^[34]. This is due to the forming and deforming of the conducting channel that occurs only at localized areas. The majority of the applied potential difference is dropped across the switching region,which is almost constant in size and independent of bulk layer thickness. Thus,the switching bias remains fairly constant with varying bulk layer thickness^{[41, 43, 98]}.

The resistivity of the HRS of MIM memristive devices increases with the thicker bulk layer due to the charge carriers needing to drift through a greater distance along the device. There are no conducting channels in the HRS of the device,thus the greater the distance,the higher the resistivity of the bulk layer. Conversely,the resistivity of LRS remains fairly constant because the device is fundamentally shorted in the LRS. This is in accordance to the published results in Reference ^[53].

In TaO₂ devices,the substrate is composed of two layers,the bulk layer with the majority TaO₂ phase and the insulator layer with the majority Ta₂O₅ phase. Because there are two layers with different phases of Ta-O,two types of comparison are performed: (i) devices with same layer ratio but different substrate thickness,and (ii) devices with different layer ratios.

7.1 TaO₂ devices with similar layer ratio

TaO₂ devices with different substrate thickness but constant thickness ratio between TaO₂ and Ta₂O₅ layers are discussed here. It was observed that the trend is similar to that of TiO₂ devices and the switching bias is fairly constant for a constant layer ratio between layers^[102]. The reason for this trend is due to the similar characteristics of HRS and LRS of TaO₂ and TiO₂ devices.

7.2 TaO₂ devices with different layer ratio

For TaO₂ devices with different layer ratios^[28],there are two types of comparison: (i) different TaO₂ bulk layer thickness with constant Ta₂O₅ insulator layer,and (ii) different Ta₂O₅ layer thickness with constant TaO₂ bulk layer.

Varying TaO₂ layer thickness (fixed Ta₂O₅ layer thickness) does not change the amount of current flow in the LRS due to the device being shorted in LRS. In the HRS,it is observed that the current flow is less in the device with the thicker TaO₂ layer 10^-8A compared to 10^-6A),because a longer device increases the overall device resistance. Due to switching regions located inside the Ta₂O₅ insulator layer,thus varying the TaO₂ layer does not affect the switching-ON bias (-1.5 V). However,the switching-OFF bias is larger with the thicker TaO₂ layer. This is due to the longer distance that the oxygen ions need to travel in the thicker TaO₂ layer. Since the Ta₂O₅ layer is constant and assuming the switching region is at fixed locations,this makes the drifting of oxygen ions from the TaO₂ layer towards the Ta₂O₅ layer more difficult due to the longer distance. Therefore,a larger switching-OFF bias is required to collapse the conducting channel,and it is concluded that varying the TaO₂ layer affects the HRS but not the LRS of the device^[28].

Varying the Ta₂O₅ layer thickness (fixed TaO₂ layer thickness) changes the current flow in the LRS and HRS of the device. Increasing the thickness of the Ta₂O₅ layer decreases the amount of current flow through the device in both LRS and HRS. Although it is documented that the device is short-circuited-like in the LRS,it is not entirely true within the Ta₂O₅ layer. This is because the MISM structure gives a similar effect to that of a Schottky barrier. This is also mathematically shown in Equation (9) that current density J is inversely proportional to insulator layer thickness d. Another observation is that both switching-ON and switching-OFF biases remain constant with different Ta₂O₅ thickness. Since the thickness of the TaO₂ bulk layer is fixed,the distance between the switching regions in the Ta₂O₅ layer to the TaO₂ layer is constant. Therefore,the switching-ON and switching-OFF biases are constant due to a similar amount of electric field being applied across the device^[28]. It is concluded that the Ta₂O₅ layer affects the amount of current in the LRS and HRS of the devices. This conclusion is also based on the fact that the thickness of the Ta₂O₅ layer is the dominating factor in manipulating the device resistance ratio^[26].

7.3 Device size of memristive devices

According to Ohm's law,the resistance of a conductor is inversely proportional to the cross-section area of the conductor. Increasing the memristive device size (defined by the cross-section area of electrodes) would decrease R_OFF only. In the HRS of the devices,there are no conducting channels,thus the electric field is applied across the entire cross-section of the bulk layer. Hence,R_OFF decreases with increasing device size.

Increasing the device size of memristive devices does not linearly decrease $R_{\mathrm{ON of TiO₂ and TaO₂ devices^[26]. When a memristive device is in LRS,it forms a thin conducting channel with the diameter much smaller than the electrode cross-section area. The conducting channels also do not span across the entire electrode^[103]. This observation was also reported for other memristive devices with similar device

structures^{[14, 104]}. Increasing device size does not necessarily mean that the conducting channels increase in size. However,it increases the probability of more conducting channels being formed^[41],but not necessarily all conducting channels are switched on when the memristive device is switched ON. Thus,since larger devices do not contribute to a large increase in conductivity,then it would be a waste to fabricate larger devices. In summary,switching parameters may be affected by device size,but the switching mechanism remains the same and overall the memristive behavior is still exhibited,which means memristive devices are scalable devices.

7.4 Electrode material selection

Electrode material does not take place in the switching process,but it affects the manner chemical reactions taking place in the bulk layer^[105]. Using the same bulk layer material,it is possible to fabricate either a unipolar resistive switching device or a bipolar resistive switching device by using different electrode materials^{[106, 107]}.

For TiO₂ devices,one of the electrodes may be replaced with other materials that still allow bipolar resistive switching to take place,such as: silver,Ag^[86],aluminum,Al^[108],and tungsten,W^[109]. For TaO₂ devices,other electrode materials that allow bipolar resistive switching to take place are such as: Al^[110],copper,Cu^[19],nickel,Ni^[111],and W^[112]. Among the electrodes used to fabricate TiO₂ and TaO₂ devices,a common trend could not be found and the effect on the performance of the devices varies,albeit several similar electrode materials used to fabricate TiO₂ and TaO₂ devices.

Electrode material selection is vital to enable resistive switching to take place in a device^[113]. Resistive switching cannot take place in a device if both top and bottom electrodes create ohmic contacts with the bulk layer. Resistive switching can only take place when at least one of the electrodes creates a Schottky barrier with the bulk layer. This also explains why certain electrode materials can be used to fabricate memristive devices with different bulk layer materials,such as tungsten,W which was used as the electrode material to fabricate memristive devices with bulk layers of TiO₂^[109],TaO₂^[112],silicon oxide,SiO₂^[114],and hafnium oxide,HfO₂^[115].

8. Conclusion

In this article,a review on the comparison between TiO₂ and TaO₂ memristive devices was presented. The comparison showed that TaO₂ memristive devices have shown better electrical properties than TiO₂ memristive devices that are of a similar physical size. This is due to the following material factors: (1) oxygen vacancies charge carrier mobility in TaO₂ is higher than that in TiO₂,(2) fewer stable Magnéli phases in the Ta-O Magnéli series than in the Ti-O Magnéli series,and (iii) smaller Gibbs free energy of the formation of TaO₂.

For memristive devices with similar dimensions and voltage applied,the switching speed of TaO₂ memristive devices is faster than TiO₂ memristive devices because the charge carrier mobility in TaO₂ is higher than that in TiO₂. Electroformation energy requirements for TaO₂ devices are also lesser due to the smaller Gibbs free energy of formation. TaO₂ devices have demonstrated memristive behavior without the electroformation process,while TiO₂ devices require an electroformation process unless additional fabrication steps are taken. Steps such as thinning the bulk layer or fabricating in high-pressure oxygen ambient allows MIM devices to exhibit memristive behavior without electroformation.

The retention capability of memristive devices has been predicted to exceed 10 years. Using temperature-accelerated degradation and the Arrhenius plot,TaO₂ memristive devices exhibit good retention past 10 years at room temperatures. Although TiO₂ devices exhibit excellent retention properties at initial stages,no temperature-accelerated degradation experiment was performed to provide such results. The endurance of memristive devices can be improved by reducing electroformation effects or using a bulk layer material that has fewer stable Magnéli phases,such as the Ta-O Magnéli series. The electrical properties of both TiO₂ and TaO₂ devices show similar scalability on changing device dimensions. However,memristive devices are usually thinner than 10 nm because increasing the device size does not contribute much to an improvement of electrical properties.

This article concludes that lesser Magnéli phases contribute to higher endurance,smaller Gibbs free energy of formation leads to smaller electroformation energies,and faster oxygen vacancies mobility provides faster resistance switching of memristive devices. For memristive devices to replace current memory applications,electrical characteristics should be calculable pre-fabrication. At the moment,accurate electrical properties of most memristive devices are only obtainable post-fabrication. Moving forward,memristive devices have great potential in increasing the density of nanoscale devices,but more research is needed to ensure a more stable resistance switching and increased endurance performance.