Corresponding author: Oleg A. Ruban ( myx.05@mail.ru ) © 2021 Andrey N. Aleshin, Nikolay V. Zenchenko, Oleg A. Ruban.
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Citation:
Aleshin AN, Zenchenko NV, Ruban OA (2021) Simulation of TiN/HfO2/Pt memristor I–V curve for different conductive filament thickness. Modern Electronic Materials 7(2): 45-51. https://doi.org/10.3897/j.moem.7.2.73289
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The operation of the TiN/HfO2/Pt bipolar memristor has been simulated by the finite elements method using the Maxwell steady state equations as a mathematical basis. The simulation provided knowledge of the effect of conductive filament thickness on the shape of the I–V curve. The conductive filament has been considered as the highly conductive Hf ion enriched HfOx phase (x < 2) whose structure is similar to a Magneli phase. In this work a mechanism has been developed describing the formation, growth and dissolution of the HfOx phase in bipolar mode of memristor operation which provides for oxygen vacancy flux control. The conductive filament has a cylindrical shape with the radius varying within 5–10 nm. An increase in the thickness of the conductive filament leads to an increase in the area of the hysteresis loop of the I–V curve due to an increase in the energy output during memristor operation. A model has been developed which allows quantitative calculations and hence can be used for the design of bipolar memristors and assessment of memristor heat loss during operation.
Maxwell equations, finite elements method, bipolar mode, conductive phase, heat loss
Researchers currently show strong interest to new computer technologies such as quantum computers and neuromorphic systems. A neuromorphic system is an artificial object imitating the work of human brain. The operation principle of neuromorphic systems is “memorizing” of new information by changing the conductivity of the contacts between artificial neurons (synapses). One possible option to implement this system is a memristor array. A memristor is a device having two functional electrodes. During memristor operation its top electrode is fed with direct voltage of different sign while the bottom electrode is earthed. Transition metal oxides e.g. TiO2, HfO2, NiO, Ta2O5 are typically used for the fabrication of the memristor working bodies. After the voltage is switched off the memristor does not change its state thus memorizing the last resistance value. During memristor operation its operation mode switches between the high resistance state (HRS) and the low resistance state (LRS). Memristor operation mode switching is achieved due to the formation and destruction of conductive filaments in the memristor working body. These filaments are high conductivity regions in the form of either clusters of positively charged oxygen vacancies with specific charge transfer mechanisms [
The I–V curves of bipolar memristors have hysteresis loops enabling the use of these electric devices as resistive memory cells. Different I–V curve branches correspond to different memristor operation modes, i.e., LRS and HRS. Local morphological changes occurring in TiO2 based memristors due to electroforming were studied earlier using atomic force microscopy [
The aim of this work is to study the effect of conductive filament thickness on the shape of the I–V curve for a TiN/HfO2/Pt bipolar memristor by numerical simulation of memristor operation mode with the finite elements method using the Maxwell steady state equations as a mathematical basis. This approach can be referred to as “first principle” simulation of I–V curves. The Hf enriched HfOx phase (x < 2) was considered to be the conductive filament. Hafnium oxide is widely used for the fabrication of bipolar memristors in which unlike titanium oxide based memristors a wider range of electrode pairs are used, i.e., Hf–TiN [
The Maxwell steady state equations have the following form:
i = σE; div i = 0; E = –grad ϕ; div E = ρ/εε0, (1)
where i is the electric current density, σ is the electrical conductivity, Е is the electric field, φ is the electric potential, ε0 is the electric constant, ε is the relative dielectric constant and ρ is the electric charge density. The Maxwell equations allow one to calculate the electric current I in the memristor as a function of the voltage U fed to the memristor top electrode for different conductive filament heights. During the simulation the memristor was considered as a capacitor consisting of two electrodes with a HfO2 layer located between them.
The memristor model was constructed in a cylindrical coordinate system (Fig.
Figure
Schematic diagram of finite elements model in cylindrical coordinate system: (a) start of conductive filament formation; (b) complete conductive filament. rf, and rm are conductive filament and memristor radii, respectively.
The material of the top memristor electrode was platinum Pt and that of the bottom electrode was titanium nitride TiN. The choice of platinum for the top electrode was stipulated by its unique properties. Under different conditions platinum can either block oxygen ions [
where Oxo is the site oxygen anion (in accordance with Kröger’s representations [
An important property of Pt stipulating its use as the hafnium oxide based memristor top electrode is the ability to exhibit catalytic properties, i.e., decomposition, by chemisorption, of gas molecules adsorbed on its surface. Earlier [
where Vad is the vacant adsorption site in platinum and Oad is the adsorbed neutral atom (adatom) of oxygen. As a result of oxygen adatom diffusion into the platinum electrode (occurring predominantly by grain boundaries) a vacant adsorption site is left on the platinum surface. On its path the oxygen adatom traps electrons from the platinum conduction band, acquires a negative charge and then as a result of recombination with the positively charged oxygen vacancy in the surface region becomes a neutral site anion. The reaction describing this process is as follows [
Thus the catalytic activity of Pt causes secondary oxidation of hafnium oxide, i.e., substitution of oxygen vacancies formed by reaction (2) for oxygen ions delivered from the environment. The process of secondary oxidation is illustrated in Fig.
The crystallographic structure of the HfOx phase was not considered in the cited earlier work [
– direct substitution of part of oxygen ions for hafnium ions;
– increase in the number of oxygen vacancies.
In the model used it was assumed that during the formation of the HfOx phase, the enrichment of the initial HfO2 lattice with Hf ions occurs by the second scheme. The mechanism of oxygen vacancy formation in an ionic crystal lattice is described by the following reaction:
This reaction results in the delivery of two free electrons to the conduction band thus increasing the metallic type conductivity in the HfOx phase forming during memristor operation. If the number of oxygen vacancies is sufficient and the vacancy subsystem has an ordered structure (this is a distinctive feature of the Magneli phase TinO2n-1 [
The memristor operation was controlled by a bipolar signal having a triangular shape (Fig.
Time-deconvoluted triangular shape bipolar signal with period τ fed to memristor top electrode for I-V curve simulation. I–IV are different sections of signal corresponding to positive (I and IV) and negative (II and III) signal slopes.
The suggested mechanism of memristor operation switching to the LRS mode based on the formation and growth of the conductive HfOx phase is a heterogeneous process that includes multiple stages which may occur either in sequence or simultaneously [
We accept that the formation of the HfOx phase in the cathode region occurs by a first order chemical reaction, i.e., HfO2 → HfOx, whose rate ω2 obeys the equation ω2 = kcx where k is the reaction rate constant ([k] = с–1) and cx is the equilibrium concentration of oxygen vacancies in the phase HfOx expressed in molar fractions. Since the formation of the HfOx phase causes consumption of oxygen vacancies, a necessary condition for the HfO2 → HfOx reaction to start is that the inequality c0 > cx is met, where c0 is the concentration of oxygen vacancies far from the interphase boundary. This condition provides for the concept of a narrow boundary layer adjacent to the interphase boundary, the rate of material supply in this layer being described by the mass transport coefficient β ([β] = с–1). The rate of material supply ω1 in the boundary layer is described by the expression ω1 = β(c0 – cx). If the supply of vacancies to the interphase boundary is the limiting stage of the process (β << k), then for the steady state case (ω1 = ω2) ω2 = βc0 [
The general expression of charged vacancy drift velocity v in the electric field Е is as follows [
where a is the crystal lattice spacing, Dv is the oxygen vacancy diffusion coefficient, q is the oxygen vacancy charge, kB is the Boltzmann constant and T is the Kelvin temperature. Introducing the characteristic electric field magnitude E0 = 2kBT/qa and using for a vacancy drifting in the electric field such a kinetic constant as the mobility mv (mv = qDv/kBT), one can rewrite Eq. (6) in the following form
which is suitable for simulation. For the migration of oxygen vacancies in hafnium oxide at room temperature (T = 300 K) E0 = 5 . 107 V/m which for the memristor model used corresponds to the voltage U0 = 0.25 V. E0 is typically used for defining the electric fields at which an electric device operates. The magnitude of weak electric fields E is far lower than E0 (E << E0) whereas fields for which the ratio E ≈ E0 is met should be considered strong [
The expression of the HfOx phase layer height as a function of time (h = vt) is true for constant voltage U. If h is represented as the two-parameter function h = h (U, t) (which is the case in this simulation), taking into account that y = sinh (U/U0) is an exponential-type function, the expression h = h (U, t) can be represented as follows:
where K0 is a semi-empirical constant which takes into account the typical duration of the memristor operation cycle. The use of Eq. (8) for I–V curve calculation allows one to exclude the time parameter from problem consideration.
The typical voltage of switching to LRS mode for bipolar memristors is 1.0–1.5 V [3, 9–12]. In this simulation the switching voltage is limited to 1.0 V. Substitution of h = 5 nm and U = 1.0 V into Eq. (8) yields K0 = 0.183 nm. This K0 was used for the calculation of the current I at all U and for any linear memristor dimensions (with the accepted limitations). Data on the electric and dielectric properties of the materials (TiN, Pt, HfO2 и HfOx) which determine the design of the memristor [
Material | Electrical conductivity (S/m) | Specific dielectric constant |
TiN | 106 | –106 |
HfO2 | 9 | 25 |
HfOx | 2 · 104 * | –106 |
Pt | 5 · 106 | –106 |
The operation of the TiN/HfO2/Pt bipolar memristor was simulated by the finite elements method using the Maxwell steady state equations as a mathematical basis. The conductive filament was the HfOx phase possessing metallic conductivity type. The memristor operation mode included four sequential time intervals corresponding to different sections of the bipolar signal having a triangular shape. Simulation was carried out for the conductive filament of a cylindrical shape whose radius was varying within 5–10 nm. The simulated I–V curves of the memristor had hysteresis loops which fact agrees with earlier experimental data. Greater conductive filament thickness corresponded to wider hysteresis loops testifying to a higher energy output during memristor operation. At sections I and IV of the bipolar signal the I (U) dependence was exponential while at Sections II and III Ohm’s law was obeyed. Depending on conductive filament thickness the memristor current varied in the same voltage range not only in LRS mode but also in HRS one. The I–V curve calculation method described in this work can be used for analyzing heat loss during memristor operation.
The work was carried out with financial support from the Russian Basic Research Fund, Grant No. 19-29-03003 MK.