Tunable Non-Volatile Memory by Conductive Ferroelectric Domain Walls in Lithium Niobate Thin Films

: Ferroelectric domain wall conductance is a rapidly growing ﬁeld. Thin-ﬁlm lithium niobate, as in lithium niobate on insulators (LNOI), appears to be an ideal template, which is tuned by the inclination of the domain wall. Thus, the precise tuning of domain wall inclination with the applied voltage can be used in non-volatile memories, which store more than binary information. In this study, we present the realization of this concept for non-volatile memories. We obtain remarkably stable set voltages by the ferroelectric nature of the device as well as a very large increase in the conduction, by at least ﬁve orders of magnitude at room temperature. Furthermore, the device conductance can be reproducibly tuned over at least two orders of magnitude. The observed domain wall (DW) conductance tunability by the applied voltage can be correlated with phase-ﬁeld simulated DW inclination evolution upon poling. Furthermore, evidence for polaron-based conduction is given.


Introduction
In recent years, increasing efforts have been made to develop novel non-volatile memory concepts to meet the increasing demands in terms of scalability and energy consumption. Conductive ferroelectric domain walls (DWs) appear an interesting approach, as ferroelectric DWs are topological defects on the atomic scale and can be created, moved and erased solely by the application of an electric field. Following the discovery of the effect of conductance of DWs in thin-film bismuth ferrite (BFO) [1,2] similar behavior was observed in various other ferroelectric thin films such as lead-zirconate titanate (PZT) [3] and lithium niobate (LNO) [4,5].
The application to non-volatile memories lies in the contradiction in ferroelectrics. Typically, they are known for their very large bandgaps. Thus, one can observe a huge variation in the conductivity between the insulating domain and the conductive domain wall. Various explanations have been made to describe the conductivity of ferroelectric DWs, ranging from oxygen or cation accumulation at the DW to polaron or electron gas formation [6][7][8][9] In various reports, the conductivity was proven to be correlated with the charge state of the DW; i.e., DWs inclined to the polar axis showed increased conductance [10][11][12] Tuning of the DW conductance was also possible by the application of an external field, which resulted in an increase in DW inclination [13].
In this publication, we want to present that precise control over the conductance of domain walls in single-crystalline LNO thin films, thus, implicitly, the inclination angle to the polar axis, can result

Phase-Field Simulation
The formation of inclined DWs is generally encountered as the reason for conductive DWs in LNO. Yet, the stable formation of such inclined DWs is still a topic of current research. We applied phase field simulations, as they can provide further evidence for the existence and stability of CDWs in ferroelectric thin films. Phase field simulations have been used to study the domain pattern formation in many proper ferroelectrics, including BTO, BFO, and PZT [28]. In this study, we have modeled the temporal evolution of ferroelectric domains in LNO thin films in response to the electrical field created by a biased probe tip. In the simulations, the voltage is ramped up to a given bias voltage. A domain nucleus is formed (shown in Figure 2a), which grows into the singlecrystalline film. Due to the external bias field, canted polarization states with in-plane polarization components are created. Afterwards, the inverted domain reaches the rear surface and grows sideways until an equilibrium is reached. When the bias voltage is released, the switched domain relaxes. We observe a stable CDW formation with a non-zero inclination angle, dependent on the maximum bias voltage applied. Since the gradient energy coefficients, which determine the DW width and energy, are rarely reported for LNO, we use the value estimated by Scrymgeour et al. [29]. We notice that increasing the gradient coefficients leads to a disappearance of the DW inclination, suggesting that the stability of inclined DWs in LNO may be attributed to its relatively small gradient energy.
The gradient coefficients of uniaxial LNO and LTO are significantly smaller than for other perovskite ferroelectrics. Previous reports on 180°DW conductance in ferroelectric thin films supports our conjecture. For example, the reported DW inclination in as-poled LTO bulk material was apparently larger [15,30]. Still, it has to be noted that the coefficients are not well known and effects from gradient energy anisotropy and carrier generation could be significant.

Phase-Field Simulation
The formation of inclined DWs is generally encountered as the reason for conductive DWs in LNO. Yet, the stable formation of such inclined DWs is still a topic of current research. We applied phase field simulations, as they can provide further evidence for the existence and stability of CDWs in ferroelectric thin films. Phase field simulations have been used to study the domain pattern formation in many proper ferroelectrics, including BTO, BFO, and PZT [28]. In this study, we have modeled the temporal evolution of ferroelectric domains in LNO thin films in response to the electrical field created by a biased probe tip. In the simulations, the voltage is ramped up to a given bias voltage. A domain nucleus is formed (shown in Figure 2a), which grows into the single-crystalline film. Due to the external bias field, canted polarization states with in-plane polarization components are created. Afterwards, the inverted domain reaches the rear surface and grows sideways until an equilibrium is reached. When the bias voltage is released, the switched domain relaxes. We observe a stable CDW formation with a non-zero inclination angle, dependent on the maximum bias voltage applied. Since the gradient energy coefficients, which determine the DW width and energy, are rarely reported for LNO, we use the value estimated by Scrymgeour et al. [29]. We notice that increasing the gradient coefficients leads to a disappearance of the DW inclination, suggesting that the stability of inclined DWs in LNO may be attributed to its relatively small gradient energy. experimental condition, domains were written with a domain size of d = 500 nm at various tip voltages. The written domains are visualized by PFM in Figure 2e. The conductivity extracted by cAFM shows a decrease in current for larger applied writing voltage. In Figure 2f the extracted maximum current values at the DW are given. A similar behavior of the current as a function of the simulated inclination angle can be observed. This is in agreement with previous theoretical assumptions [8] that the DW conductance of inclined DWs is proportional to its inclination and follows σ = 2PS sin α.

Resistive Switching Investigations
To further analyze the properties of the conductance, I-V measurements were conducted in plate-electrode condition, schematically sketched in Figure 3a under the assumption of inclined domain wall generation at nucleus sites, which would result in strong conductance changes upon reach of the local coercive voltage. Indeed, we can observe a very strong increase over five orders of magnitude in current at a very defined set voltage Vset = 21.05 V with an accuracy of ∆Vset/Vset = 10 −3 , The gradient coefficients of uniaxial LNO and LTO are significantly smaller than for other perovskite ferroelectrics. Previous reports on 180 • DW conductance in ferroelectric thin films supports our conjecture. For example, the reported DW inclination in as-poled LTO bulk material was apparently larger [15,30]. Still, it has to be noted that the coefficients are not well known and effects from gradient energy anisotropy and carrier generation could be significant.
The evolution in DW inclination for various tip voltages is given for films of a thickness of 20 nm and 50 nm in Figure 2b,c. In all given cases, a non-zero inclination can be observed after the external bias field is completely removed. The extracted final inclination angle is plotted over the homogenized applied external field. We can observe a decrease in inclination with larger applied external field. A similar behavior in the extracted DW conductance can prove a link between the degree of inclination of a DW and its conductance. To compare the theoretical prediction with the experimental condition, domains were written with a domain size of d = 500 nm at various tip voltages. The written domains are visualized by PFM in Figure 2e. The conductivity extracted by cAFM shows a decrease in current for larger applied writing voltage. In Figure 2f the extracted maximum current values at the DW are given. A similar behavior of the current as a function of the simulated inclination angle can be observed. This is in agreement with previous theoretical assumptions [8] that the DW conductance of inclined DWs is proportional to its inclination and follows σ = 2P S sin α.

Resistive Switching Investigations
To further analyze the properties of the conductance, I-V measurements were conducted in plate-electrode condition, schematically sketched in Figure 3a under the assumption of inclined domain wall generation at nucleus sites, which would result in strong conductance changes upon reach of the Crystals 2020, 10, 804 5 of 11 local coercive voltage. Indeed, we can observe a very strong increase over five orders of magnitude in current at a very defined set voltage V set = 21.05 V with an accuracy of ∆V set /V set = 10 −3 , which is proven to be the local coercive voltage from PFM measurements. This value of V set is not only reproduced for a single device, but also for 50 individual devices on a single wafer, which clearly underlines the very precise and reproducible behavior of single-crystalline resistive switching devices. Up to a voltage of −3 V, a very symmetric current-voltage relation can be observed. Yet, for larger negative biases, the absolute value of the current saturates and is not stable anymore but reduces with time. The given cycles in Figure 3d are obtained with a cycle frequency of 1.5 mHz. The observed behavior is very similar to back-switching observed upon current injection from the top electrode at small voltages confirmed by PFM. Hence, we suppose, upon the application of a negative bias, insulating straight or tail-to-tail DWs are formed or domain inversion is invoked; thus, there is no complete conductive channel anymore, which prohibits a current flow.

Conductance Type Extraction
The conduction type was investigated by I-V measurements by linearization with typical mechanisms. In Figure 4a, the I-V measurements are plotted according to space-charge-limited current (SCLC) theory, which predicts I∼V 2 in a log-log plot. A fit is given, which shows a perfect fit over almost two voltage magnitudes. In Figure 4b the I-V curve is linearized according to Shottky thermioninc emission (STE). A good fit can be seen in a medium voltage range, but both for a small and large voltage significant deviations are visible. Similarly, in the case of Poole-Frenkel (PF) linearization given in Figure 4c, we obtain a reasonable fit for medium voltages, but discrepancies for very low and high voltages. Yet, it would be not sufficient to formally exclude these two transport mechanisms. Likewise, the linearization with Fowler-Nordheim (FN) tunneling does not give a decent fit. In the LRS we obtain a flat line. Hence, the first three transport mechanisms still have to be considered. Thus, we used the abrupt junction approximation, which assumes a pn-junction with an Before switching the resistance state, one can observe no current larger than 10 pA, which is the lower limit for current detection of the applied source-meter. Further measurements with a further electrometer revealed an even smaller upper current limit of 200 fA at a voltage of −200 V, which corresponds to a resistance of at least 1 PΩ up to a bias voltage of −200 V or an electric field of 3.4 MV/cm, which underlines that leakage is negligible in the films. This similarly holds for positive Crystals 2020, 10, 804 6 of 11 read-out voltages in the high resistance state (HRS). Hence, the HRS is expected to have a resistance of at least 25 TΩ, with the low resistance state at around 10 MΩ; hence, we observe a huge resistance change over seven orders of magnitude. By setting the switch, no overshoot is present; thus the HRS setting is self-limiting. In Figure 3e the half cycle I-V-curves are shown upon the boundary condition, the current having reduced to 10 −8 A at −210 V. We can observe the reproducible set voltage. However, below this specific voltage, a slight increase in current can be observed. In general, this current, which deviates from the first cycle, is smaller the lower the current at V = −210 V. It saturates after several cycles. Several reasons are possible, e.g., deep traps, which are incorporated into the film upon large current flow. Hence, for endurance testing high voltage treatment was kept as short as possible. In Figure 3f the endurance upon such cycling is shown. The resistance is measured after every cycle. The LRS is created on application of V set = 21.1 V. The system is released into the HRS upon application of V = −210 V. This results in a very enduring resistive switching device over at least 10 5 cycles. The states can be read out without destruction. The temporal stability of the LRS is given in Figure 3g. As is visible, the conductance is very stable over at least 10 4 s. Hence, assuming an exponential decrease and a minimum current of 80%, a stability over 10 8 s or 10 years can be predicted. Similar measurements with an AFM tip revealed the stability of the current on CDWs over at least 3 h, which is about the longest time to measure due to probe drift. The statistical results given in Figure 3h show a sufficiently high margin for the application as a nonvolatile memory.
Under the application of larger set voltages and shorter pulses being applied among domain formation, the domain wall is expected to straighten, which would result in a decrease in conductance. Indeed, this can be seen as given in Figure 3i. Particularly, the fact that stronger applied electric fields result in reduced conductivity is very unusual and supports the conductance measured to be derived by inclined domain walls.
This proof of high endurance, high on/off current switching, high retention and the tenability among programming makes charged domain walls an interesting tunable nonvolatile memory.

Conductance Type Extraction
The conduction type was investigated by I-V measurements by linearization with typical mechanisms. In Figure 4a, the I-V measurements are plotted according to space-charge-limited current (SCLC) theory, which predicts I∼V 2 in a log-log plot. A fit is given, which shows a perfect fit over almost two voltage magnitudes. In Figure 4b the I-V curve is linearized according to Shottky thermioninc emission (STE). A good fit can be seen in a medium voltage range, but both for a small and large voltage significant deviations are visible. Similarly, in the case of Poole-Frenkel (PF) linearization given in Figure 4c, we obtain a reasonable fit for medium voltages, but discrepancies for very low and high voltages. Yet, it would be not sufficient to formally exclude these two transport mechanisms. Likewise, the linearization with Fowler-Nordheim (FN) tunneling does not give a decent fit. In the LRS we obtain a flat line. Hence, the first three transport mechanisms still have to be considered. Thus, we used the abrupt junction approximation, which assumes a pn-junction with an abrupt doping profile [31] Even though the main advantage of this method is to understand impedance upon a DC bias, it is still helpful to analyze the plain DC I-V curves as dlog(I)/dV is independent on most parameters, especially the contact area; it is, hence, more robust, especially upon temperature changes, and it is possible to differentiate between FN and PF. Hence, in Figure 4d, the I-V curves in a temperature range T between 300 K and 340 K are given. Only SCLC and PF [both dlog(I)/dV ∼ V −1 ] can describe such a curve, whereas STE [dlog(I)/dV ∼ V −3/4 ] cannot describe this behavior. Yet, in the case of PF, a very strong component ∝ V −1/2 should also be present, which cannot be seen in the measurements. The I-V SCLC curve fit is almost perfect over two orders of magnitude and, thus, significantly better than those of PF and STE. τ the diffusion time and D the diffusion constant. Using a conservative value of D ∼ 10 −18 m 2 /s, which is derived from experimental values of lithium transport in LixSi [32], the estimated ionic movement over 100 ms is about ∼0.8 nm at a field of 2 × 10 7 V/m and room temperature T = 300 K, which is about three orders of magnitude smaller than the film thickness. Reported values for the ionic transport diffusion constants in LNO (e.g., Li, H, D, Na, Mg) at elevated temperatures and interpolated to room temperature are significantly smaller. Hence, we can exclude ionic current having a major share. The conduction in DWs, thus, follows the Mott-Gurney equation (Child's law) [33]: I(T) = Aeff 9εμ(T)V 2 /8d 3 with Aeff the effective contact area, d the thickness of the dielectric film, ε the static permittivity, and μ(T) the mobility of the major charge carrier. Using the current extracted from cAFM measurements, we can derive a mobility of about 5 ± 3 × 10 −2 cm 2 /Vs, which is in good agreement with previously reported macroscopic photo-induced current measurements [34]. The temperature dependence in an SCLC transport regime is solely determined by the temperature dependence of the mobility of the major charge carrier.

Temperature Dependent Conductance
The temperature dependence of the conductance in the Au/Cr/LNO/Pt stack is given in Figure  5. We observe three current regimes. In ①, for temperatures above 300 K, an activation energy of 0.63 eV was obtained; in ②, between 300 K and about 100 K, we obtain an activation energy of 0.18 eV. For temperatures below 100 K ③, we obtain an activation energy of only 0.03 eV. Very similar regimes have been proposed very recently for the lifetime of bound polaron in lightly-doped Fe:LNO via Monte-Carlo simulations [35].
The processes which are assigned to these three regimes are given as: So far, we have solely discussed electronic transport. Yet, especially, resistive switching is generally an interplay of both ionic and electronic current contributions. Hence, it is necessary to further rule out scenarios of mixed electronic-ionic conduction. We use the Nernst-Ernstein equation to analyze whether ionic conduction is a significant contribution, which is given by ∆ = τeDE/kT, with τ the diffusion time and D the diffusion constant. Using a conservative value of D ∼ 10 −18 m 2 /s, which is derived from experimental values of lithium transport in Li x Si [32], the estimated ionic movement over 100 ms is about ∼0.8 nm at a field of 2 × 10 7 V/m and room temperature T = 300 K, which is about three orders of magnitude smaller than the film thickness. Reported values for the ionic transport diffusion constants in LNO (e.g., Li, H, D, Na, Mg) at elevated temperatures and interpolated to room temperature are significantly smaller. Hence, we can exclude ionic current having a major share.
The conduction in DWs, thus, follows the Mott-Gurney equation (Child's law) [33]: I(T) = A eff 9εµ(T)V 2 /8d 3 with A eff the effective contact area, d the thickness of the dielectric film, ε the static permittivity, and µ(T) the mobility of the major charge carrier. Using the current extracted from cAFM measurements, we can derive a mobility of about 5 ± 3 × 10 −2 cm 2 /Vs, which is in good agreement with previously reported macroscopic photo-induced current measurements [34]. The temperature dependence in an SCLC transport regime is solely determined by the temperature dependence of the mobility of the major charge carrier.

Temperature Dependent Conductance
The temperature dependence of the conductance in the Au/Cr/LNO/Pt stack is given in Figure 5. We observe three current regimes. In 1 , for temperatures above 300 K, an activation energy of 0.63 eV was obtained; in 2 , between 300 K and about 100 K, we obtain an activation energy of 0.18 eV. For temperatures below 100 K 3 , we obtain an activation energy of only 0.03 eV. Very similar regimes have been proposed very recently for the lifetime of bound polaron in lightly-doped Fe:LNO via Monte-Carlo simulations [35]. ③ NbLi 4+ + FeLi 3+ → NbLi 5+ + FeLi 2+ , Ea ∼ 0.03 eV. The simulated bound polaron lifetime by Mhaouech and Guilbert is overlaid on the aforementioned temperature-dependent I-V curves given in Figure 5. Thus, we can conclude that the current through the exfoliated LNO thin film is mainly governed by an interplay of bound and free small polaron transport, depending on the given temperature. As iron impurities act as deep traps, they will inhibit any further conduction. The hopping activation energy of Fe 2+ is given to be 0.35 eV [36], hence, only a little larger than Li for bound and free polarons, but as the hopping rate follows w ∼ e −r/a ≈ e −r[Å ] , hopping transport is very unlikely for the given concentrations and even for lightly doped Fe:LNO not relevant. A conversion to bound and small polarons is unlikely due to the high binding energy of 1.22 eV.

Discussion and Conclusions
In this study, we reported the domain wall conductance by inclined domain walls in LNO thin films, which could be applied as a new multi-level non-volatile memory cell based on out-of-plane ferroelectric switching, by the precise tuning of the inclination and thus the conductance state. We investigated further the temperature dependence as well as the temporal dependence on erase voltage application and concluded that the main contribution to the current is space-charge-limited.
We obtained evidence the current to be dominated by a polaron gas formation at such inclined domain wall and confined by the band bending from the temperature dependence of the conductance. Upon local inversion, generally, charge carrier injection is discussed, even though it has been speculative for a long time. Assuming such charge carriers are injected at the DW, which are needed to compensate the inclined DW of the spike-domain-like nucleus, free charge carrierselectrons-are generated, which will condensate into similarly charged and, hence, screening electron polaronic, states inside the single crystal material. Such a transition has been observed in bulk material, e.g., by optical time-resolved spectroscopy, to happen within 200 fs [37]. Yet, at a CDW these values can be significantly larger due to the smaller energy gap between conductance band and Fermi energy. However, as the band-bending is not large enough to create a 2DEG, which would be apparent with much higher mobility conductivity, we assume a 2D polaron gas (2DPG) is obtained. Bound polarons will very rapidly be trapped by iron impurities. Upon pure-diffusion transport, thus without an external electric field, the average mean free path of the bound polaron until trapping with FeLi is < tNbLi4+ ≥ 100 Å and for the free polaron is < tNbNb4+ ≥ 80 Å . These values were derived The processes which are assigned to these three regimes are given as: The simulated bound polaron lifetime by Mhaouech and Guilbert is overlaid on the aforementioned temperature-dependent I-V curves given in Figure 5.
Thus, we can conclude that the current through the exfoliated LNO thin film is mainly governed by an interplay of bound and free small polaron transport, depending on the given temperature. As iron impurities act as deep traps, they will inhibit any further conduction. The hopping activation energy of Fe 2+ is given to be 0.35 eV [36], hence, only a little larger than Li for bound and free polarons, but as the hopping rate follows w ∼ e −r/a ≈ e −r[Å] , hopping transport is very unlikely for the given concentrations and even for lightly doped Fe:LNO not relevant. A conversion to bound and small polarons is unlikely due to the high binding energy of 1.22 eV.

Discussion and Conclusions
In this study, we reported the domain wall conductance by inclined domain walls in LNO thin films, which could be applied as a new multi-level non-volatile memory cell based on out-of-plane ferroelectric switching, by the precise tuning of the inclination and thus the conductance state. We investigated further the temperature dependence as well as the temporal dependence on erase voltage application and concluded that the main contribution to the current is space-charge-limited.
We obtained evidence the current to be dominated by a polaron gas formation at such inclined domain wall and confined by the band bending from the temperature dependence of the conductance. Upon local inversion, generally, charge carrier injection is discussed, even though it has been speculative for a long time. Assuming such charge carriers are injected at the DW, which are needed to compensate the inclined DW of the spike-domain-like nucleus, free charge carriers-electrons-are generated, which will condensate into similarly charged and, hence, screening electron polaronic, states inside the single crystal material. Such a transition has been observed in bulk material, e.g., by optical time-resolved spectroscopy, to happen within 200 fs [37]. Yet, at a CDW these values can be significantly larger due to the smaller energy gap between conductance band and Fermi energy. However, as the band-bending is not large enough to create a 2DEG, which would be apparent with much higher mobility conductivity, we assume a 2D polaron gas (2DPG) is obtained. Bound polarons will very rapidly be trapped by iron impurities. Upon pure-diffusion transport, thus without an external electric field, the average mean free path of the bound polaron until trapping with Fe Li is < t NbLi4+ ≥ 100 Å and for the free polaron is < t NbNb4+ ≥ 80 Å. These values were derived by interpolation from the data of Mhaouech and Guilbert [35] under consideration of the smaller iron impurity concentration c Fe = 10 17 cm −3 and a niobium antisite defect concentration of c NbLi = 10 21 cm −3 . Upon drift condition, these values can be larger. Hence, the bound polaron has a larger mean free path. However, the hopping rate of bound polarons is only w NbLi4+ = 10 7 s −1 . Assuming the hopping rate to depend mainly on the nearest neighbor distance, like w~e r[Å] , we can calculate, that the hopping rate of free polarons is two orders of magnitude larger, hence w NbLi4+ = 10 9 s −1 . Thus, the electric current with free polaron transport as in Mg:LNO can be significantly larger. This again explains why, for the case of bulk material only, congruent Mg:LNO revealed a conductance upon UV illumination. For the ultrathin LNO films in use in this study, even undoped material can give measureable conductance.
The model explains why an asymmetric electrode condition can result in the efficient switching between an HRS and LRS. Upon the negative pulse application, the carrier density reduces, which destabilizes the inclined domain wall formation and finally reduces the conductance of the wall further. However, this process requires relatively high electric fields and times due to the low mobility of the charge carriers. Hence, the study suggests that Mg:LNO and thinner films can significantly improve the performance of the tunable non-volatile memory.
The thin-film lithium niobate resistive switching effect based on charged ferroelectric domain walls formation and erasure offers interesting advantages over conventional filament-based resistive switching, which can be interesting for neuromorphic computing applications.