SPICE Simulation of Quantum Transport in Al2O3/HfO2-Based Antifuse Memory Cells

This letter reports a compact SPICE model for the electron transport characteristics of Al2O3/HfO2-based nanolaminates for their use in multilevel one-time programmable (M-OTP) memories. The model comprises three simulation blocks corresponding to the electrical stimulus applied to the device, the equivalent circuit of the memory cell, and the events generator associated with the dielectric breakdown of the insulating layer. For a clear assessment of the quantum effects occurring in these structures, constant voltage stress was used as the primary electrical stimulus. The antifuse (AF) cell is represented by a combination of series and parallel resistances that account for the formation of filamentary conducting paths with quantum properties across the structure. The arrival of successive breakdown events is simulated using a power-law nonhomogeneous Poisson process. Our study indicates that a M-OTP memory device operating in the quantum regime not only is feasible but also that its stochastic features are addressable by circuit simulations.


SPICE Simulation of Quantum Transport in
Index Terms-Antifuse cell, oxide breakdown, memory, one-time programmable, quantum conductance unit.

I. INTRODUCTION
D IELECTRIC breakdown (BD) of a thin oxide layer is the physical mechanism behind the operational principle of antifuse (AF)-based one-time programmable (OTP) nonvolatile memories (NVM) [1]. OTPs are used in applications in which the memory devices do not need to be reprogrammed and are the preferred option for embedded NVM because of their compatibility with CMOS processes, low cost, and high scalability and reliability. Since the operation of AF structures does not rely on the charge storage mechanism, the state of  the device is not susceptible to hacking techniques based on modifying the supply voltage or temperature. This make these devices optimal for boot code, encryption keys and configuration parameters for analog sensor trimming and calibration. In addition, OTP technology has been demonstrated feasible not only in planar MOS devices [2] but also in FinFETs [3], [4] and junctionless gate-all-around nanowire transistors [5]. Multilevel memory cells (M-OTP) are also investigated with the aim of lowering the cost per unit of storage and increasing the data density [6]. Statistical properties of OTPs (voltage and area acceleration laws, multilevel generation) have been studied using the conventional time-dependent dielectric breakdown (TDDB) model (Weibull distribution) [7] and its extension, the time-dependent clustering (TDC) model (Burr XII distribution) [8].
In this letter, the conduction characteristics of Al 2 O 3 / HfO 2 -based multilevel AF memory cells are modeled and simulated. It is shown that the application of constant voltage stress (CVS) to the device generates successive BD events associated with the occurrence of filamentary conductive paths across the dielectric film. Each generated filament partially contributes to the total current so that stepwise I -t characteristics as those shown in Fig.1a are obtained. The current steps are exceptionally well defined and more than fifty events can be detected in each device for a long-term stress [8]. Fig.1b shows I -V curves before and after the occurrence of several BD events. Since the focus of this work is on the use of these devices as memory cells only events up to the sixth order will be considered here. Higher orders are subjected to accumulated variability and voltage-drop effects. Remarkably, it will be shown that the very first post-BD conductance levels are described by even numbers of the quantum unit G 0 = 2e 2 /h = 77.5 µS, where e is the electron charge and h the Planck's constant. The origin of this particular conductance step (2G 0 ) can be ascribed to a specific configuration of the atomic species (likely the superposition of atomic valence orbitals) that form the BD paths [9]. In the time domain, the evolution of the device resistance is modeled and simulated using a non-homogeneous Poisson process (NHPP). Both the quantised conductance levels and the arrival of BD events are implemented in the LTspice simulator [10]. Importantly, the devices are investigated from an electron transport perspective since the role played by the access transistor is not discussed here.

II. DEVICE FABRICATION AND MEASUREMENTS
Al/Al 2 O 3 /HfO 2 /Al 2 O 3 /HfO 2 /Al 2 O 3 /p + Si MIS capacitors with area A = 6.4·10 3 µm 2 were investigated in this work. Each dielectric layer is about 2nm-thick and they were grown by atomic layer deposition. Details about the devices and the specific combination of materials used (high diffusion barrier for Al 2 O 3 and high tunneling barrier for HfO 2 ) can be found in [11]. After measuring the fresh I -V characteristics, CVS was applied to a set of forty-one devices with −7 V applied to the Al electrode and with the Si substrate grounded. Additional experiments reveal that the generated filaments are irreversible and cannot be erased by the application of an opposite voltage.

III. BREAKDOWN MODEL FOR THE ANTIFUSE CELL
As illustrated in Fig. 2a, the proposed model for the conduction characteristics of our nanolaminates basically consists in three blocks. The first block concerns with the applied voltage V which in our case study reduces to a constant voltage source. This component can be edited so as to represent the actual signal applied to the device, for instance a voltage ramp. This signal in turn controls the BD events generator according to the implemented acceleration law (not considered here). The second block is the device itself represented by its equivalent resistance R and expressed as: where R S is the series resistance, R P the internal parallel resistance, g the conduction mode degeneracy associated with the quantum constriction, and J > 1 the number of filaments (integer number). If required, the fresh current component shown in Fig.1b can be included as a parallel leakage path. For the sake of simplicity we will focus here on the post-BD transport mode exclusively. As shown in Fig.2b, (1) in combination with the least-squares method (LSM) provides R S = 108 and R P = 1190 (41 devices × 6 levels=246 points). Fig. 2c shows that the minimum integer or half-integer value compatible with the LSM parameters is g = 2, otherwise R P < 0. Notice that (1) is a consequence of the Landauer's formula I = gG 0 V for a mesoscopic ballistic conductor [12]. Similar multiple BD events in thin oxide layers have been reported elsewhere, but mainly in connection with the resistive switching phenomenon (reversible BD) [13]. The identification of each current jump with the creation of a new filament is a consequence of the postulates of the TDC model [8], [14] which relates the BD statistics to the Poisson area scaling. Fig.2d shows that the conductance values after considering the voltage drops at R S and R P exhibit peaks at integer multiples of 2G 0 approximately. The effect fades away as the BD order increases mainly because of variability.
The third block in Fig.2a deals with the arrival of BD events and introduces the stochastic dimension in the transport characteristics. Two classes of variability need to be addressed: inherent variability associated with the random arrival of BD events (NHPP) and device-to-device (D2D) variability (likely because of oxide thickness non-uniformity [8], [15]). The generation of filaments as a function of time (t) is simulated using a power-law Poisson (PLP) process [16] (see Fig.3a): where, as shown in Fig.3b, m and n are two bivariate correlated fitting constants. Notice that for a PLP process the intensity function reads λ(t) = d /dt = m · n · t n−1 , which is consistent with the generation model of defects in high-K thin films [17]. n > 1 indicates increasing failure rate. (2) is also referred to as the Duane/Crow-AMSAA model [18]. Having found m and n for each experimental curve, the first time-to-BD ( = 1) can be calculated from (2) (see the blue line in Fig.3c). Notice the excellent correlation with the experimental first time-to-BD. Since from Fig.1b, in a first approximation, the initial leakage current as a function of the applied voltage V reads: and, according to the E-model for the voltage acceleration factor [8], the first time-to-BD can be calculated from: where a, b, c, and d are constants. Then, it can be easily demonstrated that: which establishes a connection between the initial leakage current and the first time-to-BD: larger initial currents are associated with shorter BD times as shown in Fig.3c. C and D in (5) are positive constants. In summary, the initial current value is behind the variability of n in (2) and also affects λ(t), i.e. the slope of the I-t characteristic.

IV. LTSPICE MODEL IMPLEMENTATION AND SIMULATION
In order to carry out realistic simulations, (1) and (2) must be appropriately implemented in LTspice. (1) is straightforward using a J -dependent behavioral resistance model (see Fig.4a). However, (2) involves stochasticity. Notice that (2) represents the expected failure characteristic for a single device. Since a NHPP is assumed here as a first approximation, the probability for the occurrence of a new BD event is simply given by the Poisson rule: where t is the time interval considered and λ the intensity of the process. This generation rule is expressed in LTspice as: where int is the integer part of the number, idt the time integral, ddt the time derivative, rand the random number generator, time the simulation time, SR the sampling rate, and i the random number generator seed. rand(SR·time+i) in (7) generates SR uniformly distributed random numbers in the range [0, 1] per second, while i shifts the argument of the function for every simulation run. ddt( )/SR comes from the PLP process (6). When the logic expression corresponding to the inequality is true, a unity is added to the integral counter given by idt. Now this counter value times SR scales the number of events to the appropriate magnitude. Finally, int provides the integer number of filaments J required in (1). At the onset of every simulation run, m is calculated from: where n is sampled from a lognormal distribution with parameters µ = 1.26 and σ = 0.31 (calculated using the fitdistrplus package for the R language [19]). A = 20.4 and B = 4.57 are the fitting constants reported in Fig.3b. Figs.4a and 4b show the model script in LTspice and simulations results, respectively. Take into account that, for the sake of simplicity, variability in R S , R P , and g (see Fig.4c) were not included in the above treatment.
V. CONCLUSION A SPICE model for the conduction characteristics of Al 2 O 3 /HfO 2 -based anti-fuse memory cells was reported. The model accounts for the stochastic nature of the BD phenomenon as well as the device-to-device variability. The quantum aspect of the electron transport mechanism provides a natural way to multilevel operation of OTPs.