Numerical study of synaptic behavior in amorphous HfO₂-based ferroelectric-like FETs generated by voltage-driven ion migration

The nascent era of artificial intelligence (AI) demands the processing and storage of huge amounts of data, which, from the point of view of energetic efficiency, can hardly be supported by traditional von-Neumann architectures.^1,2 Consequently, there is a strong need for finding new physical implementations of artificial neural networks (ANNs), more alike to the human brain processing schemes.^3,4 The unit elements of such ANNs should resemble the operation of the biological synapses, characterized by a hysteretic behavior with ability to alter its strength in response to applied stimuli, the so-called plasticity.^5–7 Different device architectures have been proposed to this purpose, including prevailing resistive random access memories (RRAMs) based on the formation of conductive filaments,^8–10 as well as less conventional options using polymeric electrolyte based devices,¹¹ redox,¹² or ferroelectric transistors.^11,13–15

Among these competitors to mainstream RRAMs, ferroelectric field-effect transistors (FeFETs) have already exhibited lower power consumption and faster operation,¹⁶ as well as a high endurance and retention time.^17,18 The memory and logic functions of FeFETs can be physically implemented in distinct ways. Most novel FeFETs adjust to a 1-transistor (1T) in-memory computing structure, with the ferroelectricity built-in the semiconductor channel.¹⁹ However, charge trapping at the semiconductor interface and leakage currents result in short retention times, a drawback that needs to be addressed before reaching the market.¹⁴ Nevertheless, the most common structure resembles the traditional 1-transistor-1-capacitor (1T1C) architecture, where the ferroelectric effect is provided by a dielectric oxide with a polycrystalline structure, such as HfO $2$ or ZrO $2$⁠.^20,21 Unfortunately, FeFETs based on nanocrystal ZrO $2$ are not compatible with current CMOS technology,²² while pollycrystalline doped-HfO $2$-based FeFETs are prone to suffer from high-power consumption due to undesirable leakage currents and require annealing processes up to 500  $°$C to form orthorhombic crystal phases.²³ 

A potential alternative to the material stack of FeFET (while keeping the principle of operation) is represented by ferroelectric-like FETs, which make use of the memory behavior arising from voltage-driven oxygen vacancy and negative ion migration in amorphous oxides.²⁴ The resulting switchable polarization states of the oxide (i.e., the alternate migrations of anions and cations across the oxide thickness) can be induced by applying voltage pulses to the gate contact, originating the consequent modulation of the channel conductance.^25,26 This phenomenon, which has been confirmed in experimental realizations,²⁷ enables the emulation of input pulse dependent synaptic functions,²⁸ making ferroelectric-like FETs suitable for the implementation of spiking neural networks (SNNs).¹ Moreover, devices based on amorphous dielectrics, such as amorphous HfO $2$ or Al $2$O $3$⁠,²⁹ show a better compatibility with CMOS technology than polycrystalline HfO $2$-based FeFETs,³⁰ and require a lower operation voltage,³¹ similar to the action potentials of bio-synapses.

The understanding of the physical mechanism controlling the kinetics of the involved ions is paramount to the future progress and eventual adoption of ferroelectric-like technologies in SNN. In spite of some experimental,^22,24 and exploratory numerical studies,^27,32–35 there is still a strong need of deeper comprehension so to unravel the optimal design and operation of these devices. This work presents a numerical study of the physical phenomena and operating principles in ferroelectric-like FETs driven by the redistribution of ions inside amorphous oxides. More precisely, starting by an experimental validation that accounts for the suitability of the in-house numerical tool, we assess the impact of the oxide-ion mobility and concentration, as well as of the input pulses in the response of a HfO $2$/Ge-based ferroelectric-like FET. The dependence of various electrical figures-of-merit on the physical parameters of the FET is analyzed to elucidate their optimal configuration.

The device considered in the study is schematically depicted in Fig. 1. A germanium channel with thickness t_Ge is sandwiched between two HfO $2$ regions, with thicknesses t_tox (top amorphous layer) and t_box (bottom layer). The channel length is denoted by L_ch and the source and drain regions lengths are L_s and L_d, respectively. A source/drain extension region of length L_ex is also included.

The employed SG method works under two main assumptions: (i) The applied electric fields are low enough to allow the use of the Einstein’s relation, and (ii) the discretization mesh is fine enough to assume linear variations of the electrostatic potential between adjacent positions.

Departing from an equilibrium scenario, all physical magnitudes are calculated for successive time iterations using the results of the previous time instant. For each time value, an iterative scheme is employed to self-consistently solve the Poisson-continuity system of equations.

The implementation of specific details of the numerical simulator describing the mesh employed, the boundary conditions, the program workflow or the Scharfetter–Gummel method can be found in Secs. 1–6 of the supplementary material.

Amorphous HfO $2$ is known to have a high intrinsic concentration of negatively charged oxygen ions (O^2-) and positively charged oxygen vacancies (V_O²⁺),³⁸ both of which are considered as mobile charges in the simulator. Here, the term vacancy is used to refer to the cation (V_O²⁺) left behind when the oxide is ionized by emitting an oxygen anion (O^2-). Note that both ions and vacancies are confined inside the top amorphous HfO₂ layer, i.e., they can drift and diffuse inside that material region but are not allowed to penetrate the interfaces.

It is worth to note that the ion description at the material level could involve complex chemical processes such as redox reactions, generation/recombination of ionic species, or the formation of new material phases within the device.

The DD model shows limitations to provide a comprehensive description of highly localized and non-linear processes, such as these. Nevertheless, by using macroscopic quantities like the diffusion coefficient, D_i, it can still effectively describe ion migration under electric fields, capturing transient behaviors up to reaching the steady-state, which is the aim pursued here, as reported by a number of numerical works employing a drift-diffusion scheme to describe the transport of charged ions, for example, in two-dimensional materials³⁹ or perovskites.⁴⁰ 

In order to validate the numerical simulator, its results are compared with the experimental data presented in Ref. 31. The fabricated device consists of a Ge-based p-type MOSFET with L_ch = 3  $μ$m and an amorphous HfO $2$ gate insulator with t_tox = 3 nm, exhibiting a noticeable synaptic behavior. Acceptor and donor doping of N $A= 10 17$ cm $− 3$ and N $D= 10 12$ cm $− 3$ are set in the source/drain and channel regions, respectively. The finite differences method was used to discretize Eqs. (1)–(3). In order to avoid an unsuitably large number of grid points and to minimize the computational burden, the channel length of the simulated device was scaled down by a factor $1/6$⁠, reducing it from L_ch = 3  $μ$m to L_ch = 500 nm. Exploiting the direct proportionality of the MOSFET drain current to the ratio of electron mobility to channel length, the reduction in L_ch was compensated by scaling $μ n , p$ by the same factor. Then, the mobility was adjusted to match the experimental value of the saturation current, $μ$_n = 144 cm $2$/V s and $μ$_p = 70 cm $2$/V s. The value of the dielectric constant of Ge and HfO $2$ are fixed to $ε$_Ge = 16 $ε 0$⁴¹ and $ε$_HfO2 = 20 $ε 0$⁠,⁴² respectively. The metallic contacts are modeled as ohmic, and the difference between their metal workfunction, $ϕ m$⁠, and the semiconductor electron affinity, $χ sc$⁠, is determined by shifting the threshold voltage of the transfer characteristics, V_T, obtaining a value of $χ sc− ϕ m=−0.206$ eV. The values of L_s/d and L_ex are set to $20$ and $40$ nm, respectively.

The transfer characteristics (I_ds–V_gs) depicted in Fig. 2 show very good agreement between simulations and experimental data, validating the suitability of the simulator to perform the theoretical study. Note that the leakage currents such as junction leakage and gate-induced drain leakage (GIDL),⁴¹ which result in the current saturation or slight increases in the experimental characteristics, are excluded from the simulation as they do not impact the understanding of the ferroelectric-like device operation in the region of interest. To achieve these results, a square pulse is first applied at the gate terminal with amplitude V_gs = $±$3 V and width t_w = 1  $μ$s, at a fixed drain voltage V_ds = 0 V. The application of a positive and negative gate pulse, respectively, brings the device to a high-V_T state (SET) or to a low-V_T state (RESET), caused by the redistribution of ions within the amorphous gate oxide. The SET/RESET process is followed by a linear sweep of V_gs, with a lower amplitude, enough to read the changed conductivity of the FET without perturbing the ion distribution. During the read operation, V_ds is set to $−$50 mV. The same biasing scheme and voltages are henceforth used unless otherwise mentioned explicitly.

The values $μ i= 10 − 9$ cm $2$/(V s), and c $i=2.5× 10 20$ cm $− 3$ are found by fitting the experimental results. Initially, the ion concentration is uniformly distributed in the 3-nm thick HfO₂. Some experimental works have reported that a higher ion density can be measured at the metal/oxide interface than at the oxide/semiconductor interface.^23,27,34 Nevertheless, given the thin insulating layer and the operating frequencies of the applied signals, this initial scenario would have negligible effect on the later self-consistent redistribution of ions across the insulating layer. Regarding the ion mobility, it has been proven that the mobility of positively charged oxygen vacancies is lower than that of the negative oxygen ions,⁴³ hence why some authors consider oxygen vacancies to be stationary.³³ Nonetheless, several simulation works ignore this difference between ion mobilities due to its negligible effect on the results.^24,35,44 Thus, we proceeded under the latter assumption.

The Ge density of states, g(E), is computed making use of the density functional theory (DFT) (see Sec. 8 of the supplementary material). Additionally, some works have shown that the trapping/de-trapping of carriers at the interfaces could be responsible for the clockwise hysteretic behavior when operating at frequencies up to f $≈1$ kHz.⁴⁵ This phenomenon was considered by including the traps at the HfO $2$/Ge interface. However, since the input signal here is in the MHz range,³¹ the traps were found to be not contributing to the observed hysteresis and, therefore, later removed from the simulations.

Next, the operation of ferroelectric-like FETs is investigated in detail, to predict their potential performance when integrated in ANNs. The geometrical parameters henceforth used for device simulations are listed in Table I. The rest of the physical parameters used for the experimental validation remain unchanged, except for the electron and hole mobility which are now set to their conventional values, $μ n=864$ cm $2$/(V s) and $μ p=420$ cm $2$/(V s). We also assume to discard the presence of traps due to their small impact in the simulations as they do not contribute to hysteresis, but only to small adjustments on V_T. Specifically, the capture and emission time constants of the traps range from $τ e , c∈[10,100]$ s according to previous studies,⁴⁶ meaning their occupancy levels remain unchanged during the few microseconds of the voltage sweep intervals.

Figure 3(a) shows the transfer characteristics of the ferroelectric-like FET upon application of a fixed amplitude SET/RESET pulse with V_gs = $±$3 V and varying t_w. The progressive change in V_T, achieved by modifying the t_w, can be used to store multibit binary data or modulate the weight change in synaptic emulation. The current values change significantly with retention time, varying by three orders of magnitude for times ranging from 0.1 to 2  $μ$s. These current levels can be encoded into binary digits, similar to multilevel digital memories.²⁸ Additionally, the analog current modulation produced by the input pulses resemble the learning process carried out in biological synapses.⁴⁷ The maximum change in the conductivity obtained upon SET/RESET of the device can be captured by the memory window (MW), defined as MW $=$ V $T$(RESET) $−$ V $T$(SET). Note that V_T of the device is calculated at a constant current of 0.1  $μ$A/ $μ$m. Figure 3(b) proves that a larger t_w increases the resulting MW.

Shortly after the end of the SET/RESET pulse, the ions start diffusing back, giving the ferroelectric-like FET its volatile behavior. Figure 3(c) shows the I_ds time evolution after the end of a SET pulse. As shown, the current I_ds gradually decreases to its steady state value due to ions diffusing back to their equilibrium position. The retention time (t_ret) is defined as the time required for the drain current to drop to a 1 $%$ of its maximum value,⁴⁸ as follows: t $ret=Δ$t @ $Δ$I = $Δ$I_max/100. Here, the maximum drain current is the one measured right after the writing pulse of duration t_w, I_ds(t_w), and currents are referenced to the steady state value, I_ds(st.), so that $Δ$I_max = $I ds( t w)− I ds( s t .)$ and $Δ$I(t) = $I(t)− I ds( s t .)$⁠. It is noteworthy that the value of t_ret, plotted in Fig. 3(d), decreases with increasing t_w, a behavior opposite to that of MW. An alternative definition of t_ret that considers the current dropping to a 10 $%$ of $Δ$I_max is also analyzed [Fig. 3(d), dashed lines], showing the same monotonically decreasing behavior of t_ret.

To gain further insights into the opposing behavior of MW and t_ret, let us consider Fig. 4, which shows the oxygen ions and vacancies distribution, as well as the potential and electric field across the gate oxide thickness as a function of time and for different t_w. A SET pulse of V_gs = $3$ V at V_ds = 0 V is used and each row stands for a t_w value, from 500 ns to 5  $μ$s. As can be seen, both ions and vacancies are drifted by the SET pulses from the initial uniform distribution toward the interfaces of the oxide layer, with oxygen ions accumulating under the metallic contact and vacancies moving toward the channel. This redistribution, more accentuated for longer t_w values, gives rise to a higher electric field (E-field) as well. The peak absolute value of the E-field across the gate oxide is $≈6$ MV/cm and $≈7$ MV/cm for t_w = 500 ns and 5  $μ$s, respectively. Therefore, with increasing t_w, the internal E-field (in direction opposite to the externally applied E-field) also increases, further shifting the channel V_T and resulting in a wider MW. When the stimulus disappears (dashed vertical line, V_gs = 0 V after t_w), ions continue drifting. Their low mobility causes this delayed reaction and, as a consequence, the hysteretic behavior of ferroelectric-like FETs. When the drift ends, ions start diffusing back from the interfaces, eventually reaching the initial uniform distribution. It can be seen that the longer the duration of the pulse, the higher the concentration of ions at the interfaces. The maximum concentration of oxygen ions at the interfaces are found to be $≈3.9× 10 20$ cm^-3 and $≈2.43× 10 21$ cm^-3 for t_w = 500 ns and 5  $μ$s, respectively. In conclusion, the longer the t_w, the higher the drift and the built-in E-field that gives rise to a wider MW. However, at the same time, a higher t_w increases the concentration gradient, therefore increasing the diffusion and reducing the t_ret, giving rise to the aforementioned trade-off between MW and t_ret with respect to t_w.

The drift and diffusion of ions, and specially their influence on MW and t_ret, is expected to depend on the mobility (⁠ $μ i$⁠) and the concentration (c $i$⁠) of ions. Beside being appropriate for its use as synaptic element, depending on the values of the MW and t_ret, the ferroelectric-like FET could find varied applications, such as memory selector, true random number generator or in artificial neurons.⁴⁹ Therefore, the better understanding of the dependence of MW and t_ret on $μ i$ and c $i$ could provide accurate guiding principles to optimize the design of ferroelectric-like FETs for targeted applications. Several values of c_i and $μ$_i distributed around the parameters issued from the experimental validation (c_i = 2.5 $×$ 10²⁰ cm^-3 and $μ$_i = 10^-9 cm²/V s) have been tested, in particular: c_i $=[0.25,0.5,1.25,2.5,5,12.5,25]× 10 20$ cm $− 3$ and $μ$_i $=[0.1,0.2,0.5,1,2,5,10]× 10 − 9$ cm $2$/V s. Figure 5(a) shows the MW as a function of $μ i$ and c $i$ for t_w = 1  $μ$s. It can be seen that higher values of $μ i$ (resulting into higher drift velocities) and c $i$ (producing larger gradients of ion concentrations) enable larger MWs as larger E-fields can be generated within the oxide. Nevertheless, the ion concentration has an upper theoretical limit at the complete ionization of HfO₂, while ion mobility cannot be too high so to avoid changing the distribution of ions when the read signal is applied. The MW also appears to have a weaker dependence on the concentration as compared to the mobility of the ions. Similarly, Fig. 5(b) shows t_ret as a function of $μ i$ and c $i$ for t_w = 1  $μ$s. Here, t_ret decreases with increasing $μ i$ and c $i$ values, with quite similar behavior with respect to both quantities. Figures 5(a) and 5(b) also suggest an interdependence between $μ i$ and c $i$ to achieve the desired value of MW and t_ret. Figures 3(c) and 3(d) and Figs. 5(a) and 5(b) clearly depict a trade-off between the two figures-of-merit of the ferroelectric-like FETs, as illustrated more clearly in Figs. 5(c) and 5(d) with respect to c $i$ and $μ i$⁠, respectively. Therefore, a careful design is needed for a specific target application.

In addition to the V_T change achieved by varying t_w in Fig. 3(a), another way to continuously tune the V_T is by repeatedly applying SET/RESET pulses of same amplitude and t_w to emulate the paired-pulse facilitation (PPF) and paired-pulse depression (PPD) functions of the biological synapse. Figure 6 shows the variation of I_ds at read voltages V_gs = $−$1 V and V_ds = $−$50 mV, after each consecutively applied SET/RESET pulse of t_w = 1  $μ$s. The progressive application of SET pulses results into a PPD, where a change in the I_ds of about 14 orders of magnitude could be achieved. The current can then be gradually increased (PPF) by the subsequent application of a similar amount of 15 consecutive RESET pulses, returning to a high level of current. Thus, the application of the appropriate set of writing and reading pulses, with suitable amplitudes and durations, induces a synaptic-like behavior on ferroelectric-like FETs, an essential condition for their further implementation in ANNs.

The development of analytical models and numerical simulators is crucial for understanding physical mechanisms and parameters that are difficult or costly to access experimentally. This work presented an in-house numerical tool able to self-consistently simulate the electrostatic and time-dependent electronic and ionic transports in iontronic devices. Simulations are proved to successfully reproduce the experimental data of the electrical characteristics of ferroelectric-like FETs, and they can accurately explore their properties, providing valuable insights into optimizing design parameters for a better device operation. The dependence of the figures-of-merit, memory window and retention time, on the device physical parameters and their origin were analyzed. A trade-off was noticed between memory window and retention time with respect to the duration of the writing pulse, and the mobility and concentration of the ions. The change in the conductivity of the ferroelectric-like FETs in response to a train of pulses is also proved to emulate the paired-pulse facilitation and paired-pulse depression functionality of the biological synapses.

In conclusion, voltage-driven ion migration not only tunes the conductance and threshold voltage of these devices, but also opens the possibility to create energy-efficient artificial neurons out of ferroelectric-like FETs based on amorphous gate oxides. In view of such promising results, their application on ANNs should be further pursued, and their operation optimized to foster the development of the neuromorphic computing paradigm. We hope that this study will act as a benchmark for refining the fabrication process and obtaining the targeted performance.

See the supplementary material for additional information about the numerical tool implementation, elaborating the finite differences method, boundary conditions, analysis of the steady-state and time-dependent regime of charge carriers and ions, the program workflow and the Scharfetter–Gummel method. Further details about interface traps and the density of states calculation for Germanium are also provided.

This work was supported by the Spanish Government through Project Nos. PID2020-116518GB-I00 funded by MCIN/AEI/10.13039/501100011033 and TED2021-129769B-I00 funded by MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR. This work is also supported by the R+D+i Project No. A-ING-253-UGR23 AMBITIONS co-financed by Consejería de Universidad, Investigación e Innovación and the European Union under the FEDER Andalucía 2021-2027. J. Cuesta-Lopez acknowledges the FPU19/05132 program. M. D. Ganeriwala acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 101032701.

The authors have no conflicts to disclose.

J.C.-L. and A.G. are the corresponding authors. J.C.-L. and A.T.-L. developed the simulator. J.C.-L., E.G.M., F.G.R. and F.P. conceived the numerical experiments. J.C.-L. and M.D.G. conducted the simulations and data processing. E.G.M., F.G.R. and A.G. acquired funding. J.C.-L., E.G.M. and M.D.G. wrote the original draft. All authors contributed to the revision and editing of the final manuscript. All authors have read and agreed to the published version of the manuscript.

J. Cuesta-Lopez: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). M. D. Ganeriwala: Formal analysis (equal); Investigation (supporting); Methodology (supporting); Software (lead); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). E. G. Marin: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal). A. Toral-Lopez: Methodology (equal); Software (lead); Supervision (supporting); Writing – review & editing (supporting). F. Pasadas: Conceptualization (supporting); Funding acquisition (equal); Investigation (supporting); Methodology (supporting); Project administration (equal); Resources (equal); Writing – review & editing (equal). F. G. Ruiz: Conceptualization (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (supporting); Project administration (lead); Resources (lead). A. Godoy: Conceptualization (supporting); Formal analysis (supporting); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.