Linking the Intrinsic Electrical Response of Ferroelectric Devices to Material Properties by Means of Impedance Spectroscopy

Ferroelectric devices have gained attention in recent years as a potential solution for ultra-low power computing due to their ability to act as memory units and synaptic weights in brain-inspired architectures. One way to study the behavior of these devices under different conditions, particularly the influence of material composition and charge trapping on ferroelectric switching, is through impedance spectroscopy. However, the parasitic impedance of the metal lines that contact the electrodes of the device can affect the measured response and interpretation of the results. In this study, we examined the frequency response of ferroelectric tunnel junctions (FTJs) with a metal-dielectric-ferroelectric-metal (MDFM) stack at various voltages, starting from the analysis of single layer capacitors (MFM and MDM) to better interpret FTJ’s results. To accurately assess the intrinsic response of the device, we developed a method that estimates and removes the parasitic access impedance contribution, which was validated by means of physics-based simulations. This method allows quantifying the intrinsic device-level variability of FTJs and, for the first time, to investigate the relation between the thickness of the dielectric layer, the equivalent phase composition of the ferroelectric material, and the magnitude of the peak in the frequency response, often assumed to be related to charge trapping only.


I. INTRODUCTION
I N THE last decade the technological research on computer sciences and electronics promoted enormous improvements in terms of computational capability, data analysis and development of autonomous systems, with notable consequences in the advancements of Artificial Intelligence [1], Internet of Things (IoT) [2] and Big Data management [3]. However, these improvements are ever-increasingly revealing the limitations of typical CMOS computing architectures, e.g., scaling, thermal and energy efficiency constraints [4], [5], [6]. In particular, the latter is mainly due to the intrinsic separation of the processing and memory units (known as Von Neumann bottleneck (VNB)) and their constant need to communicate [7], [8]. Manuscript  Consequently, a change of paradigm is urgently needed in order to comply with the modern technological requirements in a responsible and sustainable way [9].
Among the explored solutions, some of the most promising candidates to lead this technological transition can be found in ferroelectric hafnium zirconium oxide (HZO) based devices [22], [23], [24] combining remarkable advantages, such as low power consumption, CMOS compatibility, fast access speeds, high-scalability, low footprint, and non-volatility. In particular, the devices studied in this work are ferroelectric tunnel junction (FTJ) memories consisting of a metal-dielectric (DE)-ferroelectric (FE)-metal (MDFM) stack which act as synaptic elements in neuromorphic circuits [25], [26], [27] with a non-destructive readout [28], [29]. The information is stored by means of the device remnant polarization, that can be read by sensing the leakage current upon the application of a small voltage pulse. Also, the characteristics of these devices allow for specific tailoring of the operational conditions (e.g., speed and voltage) to best suit different specific applications (e.g., microwave applications or synaptic weights in neural networks) [30], [31], highlighting the flexibility of this technological solution. However, a detailed and comprehensive electrical characterization for different conditions is needed before a dependable introduction of FTJs in actual circuits.
Impedance spectroscopy [32] is a common technique used to investigate the role of each layer (FE and DE) and charge trapping in ferroelectric switching. To estimate the trapped charge response, the G p /ω peak is often used as a parameter, where G p is the equivalent conductance in the total admittance of the device under test (Y(ω) = jωC p //G p ). This value is obtained through capacitance-frequency/conductancefrequency (C-f/G-f) measurements, in which a small-signal with a varying frequency is applied on top of a bias voltage. This technique allows for a detailed analysis of the electrical response of ferroelectric devices. However, especially when investigating lab-level samples, test devices may be strongly affected by the parasitic impedance of the metal lines (or broken access device [33], [34]) contacting the electrodes of the device, which is not always negligible and may strongly alter the measured response, with consequences on the interpretation of the results.
In this research, we extend our conference paper [35] and expand upon our previous findings on the validation of the small-signal model for ferroelectric tunnel junctions (FTJs) [33], [34], also used to investigate aging mechanisms [36], and examine the electrical response of FTJs with a metal-dielectric-ferroelectric-metal (MDFM) stack at various voltages through multi-voltage capacitancefrequency/conductance-frequency (C-f/G-f) measurements.
We compare the results of these measurements to those obtained from single layer capacitors (MFM and MDM) to gain a deeper understanding of the electrical properties of these devices. To accurately model the electrical response of the FTJs, we employed small signal models (shown in Fig. 1 a-b-c) that consider: i) the distinct leakage and capacitance paths for each layer, ii) the first order defects contribution, and iii) an equivalent parasitic impedance (Z SER ) in series with the device, which is modeled as the parallel of C SER and G SER . The extracted Z SER parameters were validated through physics-based simulation with the Ginestra R [37] simulation platform and then subtracted from the measured response to reveal the intrinsic response and device-level variability. The extracted intrinsic response is then compared with simulations of the intrinsic device to further confirm the dependability of the proposed approach and better interpret the results.
The paper is organized as follows: Section II presents the details of the performed experiments and the studied devices, together with their small-signal model. In Section III we analyze the outcomes obtained by the single-layer capacitors (MDM and MFM) characterization, the validation of the extracted Z SER , and the intrinsic device-to-device variability. In Section IV we repeat the same study on FTJs. In Section V we then compare and discuss the relation found by varying the dielectric and ferroelectric thicknesses (t DE and t FE ) in intrinsic FTJ devices and compare the results versus simulations. Conclusions follow. Fig. 2. a) Multi-voltage C-f/G-f measure starting from positive voltages (blue) and negative voltages (black). For each bias, a small ac signal (30mV) with frequencies from 1kHz to 10MHz is superimposed. b) The total measured admittance (Y(ω) = jωC p //G p ) is analysed with the models of Fig. 1a-b-c. c-d) Example of rFE (V) and G FE (V) extracted by the model for different polarizations [36], respecting the same colour code as in a).

II. DEVICE AND EXPERIMENTS
We study MFM, MDM, and FTJ frequency response by performing multi-voltage C-f/G-f measurements, as shown in Fig. 2a , superposing, for each bias, a 30mV RMS AC signal with frequency sweeping from 1kHz to 10MHz. MFM and FTJs are measured for both negative and positive voltages to check the expected relation between the model parameters (specifically the FE permittivity ( rFE ) and conductance (G FE ) as depicted in Fig. 2c-d) and the applied voltage polarity as well as the ferroelectric polarization. We also limited the bias to a safe range, in order to prevent device degradation and, ultimately, device breakdown. MDM capacitors have been measured only in a small range ([0 +1] V) since, for simplicity, the analysis and comparison with other devices will be focused only on the results obtained at 0V DC bias. However, the MDM model is able to reproduce C p and G p /ω profiles ( Fig. 3a-b) for higher bias ranges and accounts for the conductance (G DE ) voltage dependance, as reported in Fig. 3c for an MDM with t DE = 6nm. Furthermore, the small signal trap response is found to be weakly influenced by applied bias, which makes investigating large voltage ranges superfluous. Fig. 1a-b-c shows, for each device, the compact small-signal model used to map the total measured admittance (the parallel of an overall measured capacitance, C p , and conductance, G p ) to specific layer-related parameters, Fig. 2b. The models account for a capacitance and conductance path for each layer (to separately consider the leakage of each layer), and a series impedance Z SER (C SER //G SER ) to model the parasitic impedance of the access metal lines, which cannot be removed with open-circuit and/or short-circuit compensation [33], [34], which are however performed before the measurements. A C it -G it branch is also inserted between the TiN electrodes (for MFM and MDM) to model to the first  order the presence of interface defects at the parasitic TiON / TiAlO layers caused by post-deposition annealing [28], [38]. To simplify the overall FTJ model, we included only a single C it -G it branch across DE to consider the equivalent effect of all interface's defects (M-DE, DE-FE, FE-M), as they are most likely mainly located at the DE-FE interface [39], [40]. Different other attempts in positioning this branch have been tried, without meaningful and relevant results [34]. Though the model can, with no a priori constraints, reproduce the expected voltage dependence of all parameters for different polarizations (e.g., the typical butterfly-shaped ferroelectric permittivity, rFE , vs. voltage relation [38], Fig. 2b), the MFM and FTJ results are hereafter reported, for simplicity, only for positive polarization (i.e., +3(+4) V → −3(−4) V). Notice that rFE represents an effective permittivity, accounting for both orthorhombic (i.e., ferroelectric) and non-ferroelectric phases present in the FE [33], [34], [41].

III. SINGLE LAYER CAPACITORS
In order to better understand the response of materials and defects, we began by studying MFM and MDM single layer capacitors. Fig. 4 illustrates the small-signal model for MFM capacitors and the measurement results for various MFM devices at 0V in terms of C p and G p /ω, along with modeled profiles. To investigate the effect of access impedance, we measured the same device with the tip positioned at different locations on the metal pad (resulting in different current paths to the capacitor) as well as devices with different areas but with the tip position fixed on the metal pad. This allowed us to examine the electrical response of these devices and how it is influenced by various factors. Fig. 5 reports the comparison of the extracted parameters for each voltage and for each device (different symbols and colors), emphasizing a strong device-level variability especially in the series conductance (G SER ). Removing Z SER from the model and keeping the other parameters fixed, it is possible to retrieve the intrinsic MFM C p and G p /ω profiles (Fig. 5). Results show that: i) the intrinsic device-level dispersion is much smaller than what observed in Fig. 4; ii) the high-frequency C p roll-off is due to the access impedance and the intrinsic C p profile is, as expected, frequencyindependent [42]; iii) the peak in the intrinsic G p /ω profile, usually related to defects response [43], was hidden by the access impedance. The real one is much lower than that observed in Fig. 4 and occurs at lower frequencies.
To validate the extracted Z SER value we performed independent physics-based C-f/G-f simulations of a 10nm MFM stack (reported in Fig. 6a), for simplicity only at 0V. Such physicsbased simulations are carried out using Ginestra R simulation platform [37], which includes Schottky and thermionic emission, direct (WKB approximation), trap-assisted tunneling (TAT -including trap-to-trap contribution), and Fowler-Nordheim tunneling, as well as the trapped charge term in the Poisson's equation. We include defects in the HfO 2 bulk (defects density δ = 10 20 cm −3 with a normal distribution in space centered at the middle of the stack and a uniform lateral distribution) having thermal (E TH = 2.1±1 eV) and relaxation (E REL = 1.2 eV) energies [44] which are very close to those predicted by hybrid-DFT calculations for oxygen  vacancies in the orthorhombic ferroelectric phase of HfO 2 (E TH ≈ 1.8 eV, E REL ≈ 0.7 eV) [45]. Consistently with the value extracted from experiments using the small-signal compact model we imposed an equivalent rFE (0V) = 26, in order to effectively include the presence of different HZO phases (e.g., orthorhombic, monoclinic, and tetragonal).
Results are reported in Fig. 6b and show that the measured C p and G p /ω profiles (blue symbols) can be only reproduced by including a parasitic series impedance equal to the extracted Z SER (black lines), while simulations without the parasitic impedance (red dashed lines) show very similar profiles to those of the intrinsic MFM in Fig. 5, especially for the C p profile, (Fig. 6b). The simulated G p /ω profile presents high and low frequency behavior in quantitative agreement with those of the intrinsic device as retrieved by excluding the Z SER contribution from experimental data using the compact model. However, the peak amplitude and position in the mid-frequency range are slightly different. This can be explained by the simplifications adopted in simulations, i.e., we included just one defect species (oxygen vacancies) and neglected the presence of possible interfacial layers and different HZO phases, which would be needed to carefully model the complex dynamics of a real stack [46], [47], [48]. However, in this work we are mainly interested in the analysis of qualitative trends rather than in the precise and comprehensive investigation of the underlying complex phenomena.
The same experiments are also repeated for different MDM stacks (Figs. 7-8), revealing the intrinsic behavior of these capacitors, which show a defects response in the G p /ω profile at much higher frequencies compared to the MFM. Differently form Fig. 3, which reports results for MDM with t DE = 6nm, Fig. 7 shows also that 10nm MDMs present much lower (as expected) G DE s in [0 +1] V. The trend is found to be almost constant within the applied voltage range, most probably because of the noise floor limitation of the measurement setup. A G DE increase would then be visible at larger voltages, however respecting the trends in Fig. 3. As for thinner MDM, traps response is still weakly dependent on bias.

IV. FERROELECTRIC TUNNEL JUNCTIONS
In Fig. 9a-b, the measurement results of an FTJ with a t DE value of 2.5 nm at 0V are shown for devices with different areas and tip positions. By analyzing these results, it is possible to extract the intrinsic profiles, depicted in Fig. 9c-d, and parameters (shown in Fig. 9e) by plotting C p and G p /ω discounting the Z SER parameters. As expected, the G FE values, presented in Fig. 9e, are similar to those of the MFM (shown in Fig. 5), as both devices have a t FE of 10 nm. It is worth noting that the frequencies and values of the G p /ω peak in the FTJ devices are similar to those in the MFM devices, which suggests that the intrinsic response of the FTJ is more sensitive to defects in the FE rather than defects in the DE or at the FE/DE interface.
This comparison between the results obtained on FTJs and those obtained on MFM and MDM structures helps to better understand the response measured on FTJs.

V. DISCUSSION
Repeating the experiments for FTJs with different t DE it is possible to compare the extracted parameters and devices response with those of the MFM and MDM capacitors. As mentioned before, FTJ G p /ω peak's frequencies are much more similar to those of MFM capacitors (reported in Fig. 10a) rather than those of MDMs, several orders of magnitude above the others. This further confirms the role of FE properties and defects nature in determining the small-signal response for FTJ devices. Furthermore, the extracted C it values for FTJs show no trend with t DE , indicating a negligible relation between the DE-FE interface impurities and DE thickness. Fig. 11a reports the G DE vs. t DE exponential trend obtained by interpolating the values extracted for different MDM capacitors, compared with those extracted from FTJs. The latter show higher than expected values with a very mild dependence on t DE . This is also confirmed by ultra-low frequency IV measurements (execution time = 132s), Fig. 11b, which reveal that FTJs with t DE ≤ 3nm all have similar leakage. Fig. 10. a) Comparison between the extracted G p /ω peak's amplitudes for MFMs (blue), FTJs (red) and MDMs (black), revealing the similarity between FTJs and MFMs. b) Comparison of extracted C it vs. t DE in FTJs. For each t DE , the reported C it ranges consider the parameter voltage dependence and include device to device variability (as in Fig. 9). No trend is found. Fig. 11. a) GDE vs. t DE in MDMs (black) and FTJs (red). FTJs present higher than expected GDE values, highlighting the increase (and saturation (purple circle)) of DE defectivity with reducing its thickness. b) Slow IV measurements (execution time = 132s) with a reduced capacitive current (dV/dt) to highlight the leakage contribution emphasize the similarity for FTJs with t DE ≤ 3nm (purple circle), confirming the findings in (a).
This confirms that the ultra-thin DE layer in FTJs is highly defective and dominated by impurities probably out-diffusing from the interfaces with the top TiN electrode and the FE. However, t DE is found to modulate the FE properties, specifically the voltage dependence of rFE and the G p /ω peak. Fig. 12a shows that increasing t DE results in lower and more compact rFE profiles, suggesting that a thicker DE can partially inhibit the orthorhombic phase formation during the annealing process, affecting the switching. Arguably, this is a local phenomenon expected to occur close to the DE/FE interface. Furthermore, the effect of t DE is also visible in the analysis of the G p /ω peaks at 0V, as reported in Fig. 12b, suggesting an inverse relation between t DE and ferroelectric domain response. Interestingly, the extracted rFE and G p /ω peak values show a linear correlation, Fig. 12c, reinforcing the idea that the G p /ω peak is not only related to defects as usually thought [43], but is also related to the overall FE phase composition. Thus, extra care must be adopted when assessing the interface defects density based only on G p /ω peaks [43] as this can easily lead to wrong conclusions. It is therefore arguable that the dielectric thickness may have an influence, specifically during the post-deposition annealing, on the formation and physical composition of the expectedly disordered parasitic layer between the FE and DE [39], [40], [49]. Such a layer is expected to have a lower permittivity compared to that typical of ferroelectric HfO 2 , which would result in an effective change in the observed rFE parameter as extracted from our experiments. Along the same line, t DE is found to influence also the intrinsic conductance of the FE layer, G FE , especially at negative voltages, Fig. 13.
To confirm the found trend, the same MFM stack of Fig. 6a is simulated again in Ginestra R by changing the rFE value in order to mimic the effect on the FE layer resulting from the insertion of a DE layer during the fabrication process. Simulation results show an almost linear relation between rFE and G p /ω peak's amplitude (Fig. 14), qualitatively in agreement with the trend extracted from measurements and model extractions from all FTJs, as shown in Fig. 12. Moreover, by changing the rFE the low-frequency G p /ω tail changes as well, which is associated to a larger leakage, in turn dominated by the intrinsic FE conductance [33], [34], which is in agreement with the experimental results in Fig. 12c, further stressing the role of FE properties on G p /ω response [36]. Inizio moduloThen, the increase of the G p /ω peak's values would be mostly related to the FE leakage increase because of the constant defect's contribution, which can indeed be mainly associated to the frequency of the G p /ω peak. Fig. 13. Comparison of the G FE (V) extracted by using the small-signal model for the MFM capacitor and FTJs. Larger G FE is found for thinner DE layers, especially for negative voltages.  Fig. 6a. A linear trend is found between rFE and G p /ω peaks. A larger low-frequency tail is observed for higher rFE which is associated with larger FE conductance.

VI. CONCLUSION
In this work, we introduced and validated an advanced FTJ small-signal compact model that accounts for separate leakage contributions in the FE and DE layers, the contribution of a parasitic series impedance, and non-uniform crystalline FE phase. The model correctly reproduces measurements taken on different devices in different conditions and with different tips position, allowing a more refined investigation on sample layout and material properties effects on the entire device under measurement. In particular, the possibility to isolate and remove the parasitic impedance between the actual device and the bottom tip, validated by physics-based simulation in Ginestra R simulation platform [37], allows the analysis of the desired intrinsic devices properties and variability.
Results are obtained by comparing the study of single layer MFM and MDM capacitors with MDFM FTJs. The insertion of t DE in FTJs, although weakly effective in leakage control due to the found high defectivity, is revealed to be a possible cause for inhibiting the ferroelectric orthorhombic phase formation in HZO layer or, equivalently, for promoting the erosion of a portion of the FE layer resulting in the formation of a parasitic non-ferroelectric layer with different dielectric properties. The equivalent fraction of orthorhombic phase, that is strictly related to rFE , is also found to be approximately linearly related to the peak value of the G p /ω vs. frequency curve, as also confirmed by physics-based simulations. The found relation between G p /ω peak's amplitude and rFE (the changes of which are suggested to be related to t DE ), would be also explained as a direct consequence of the direct proportionality between rFE and G FE (dictating the low-frequency behavior [33], [34]), suggesting that the typically adopted estimation methods for interface trap density may be misleading.