# Toward a better understanding of RRAM

During the reset process, the probability of Vo recombination is
determined by the product of the attempt-to-escape rate and the
concentration of the oxygen ion at that position as given in equation 6:

*t*

_{0}is the lattice

*vibration time ~10*

^{-13}s,

*t*is the simulation time step,

*a*is the lattice constant (also the mesh size) ~0.25 nm,

*E*

_{a}is the Vo formation energy ~ 1 eV,

*E*

_{m}is the oxygen ion migration barrier ~1 eV,

*E*is the electric field, and

*C*is the ion concentration.

At each grid point, a random number is generated and compared with the probability calculated above to determine success of generation (if that grid has no Vo) or recombination (if that grid has Vo). In each iteration step, the electric field and temperature are updated: the voltage is assumed to be dropped on the gap region since the highly conductive filaments can be viewed as a virtual electrode [3]. For the sake of simplicity, an average temperature inside the simulated cell is calculated from the macroscopic Joule heating following the analysis in [10]. A more rigorous treatment is to calculate the local temperature profile from the Fourier heat equation by considering the power dissipation due to the inelastic tunneling [11]. Figure 6 shows the established stochastic simulation flow. To handle the current overshoot problem with a drastic increase of the Vo density during the set process, a trial-and-error strategy to optimize the simulation time step is used.

**Figure 6: The established stochastic simulation flow.**

Figure 7 shows the simulated switching I-V curves of forming/reset/set and the corresponding Vo configuration with the percolation paths highlighted. It is seen that a larger reset stop voltage would result in a larger gap thus a higher HRS. A larger set compliance would result in more percolation paths thus a lower LRS. The abrupt set transition is due to a positive feedback of the temperature and field enhancement of the Vo generation probability. The gradual reset transition is due to the oxygen ion migration from the interface and gradual recombination of Vo and consequent increase of the gap distance. For a better appreciation of the switching dynamics, an animation video showing the stochastic Vo evolution during the DC cycling is provided in [12].

*Click image to enlarge*

**Figure 7: Cross-section view of the simulated cell (left electrode: positive bias for forming/set and negative bias for reset, pink points are Vo), simulated cell (10 nm ×10 nm) corresponds to the weak spot region of a RRAM cell, e.g. the grain boundary. (a) initially randomly distributed Vo in as-fabricated cell; (b) forming process; (c) percolation paths after forming; (d) reset process with different stop voltages; (e) & (f) smaller/larger gap due to smaller/larger reset stop voltage; (g) set process with different compliance current; the current overshoot during forming/set is also shown; (h) & (i) fewer/more percolation paths due to smaller/larger compliance current. Percolation paths are found by the Dijkstra algorithm and color-coded in grey scale according to the conducting strength (darker means stronger).**

Experimental

Experimental

**corroboration**

We developed DC I-V curves for the experimental (see figure 8) and simulated cases (see figure 9).

**Figure 8: Experimental DC I-V characteristics of HfO**

_{x }memory for different reset stop voltages demonstrate abrupt set and gradual reset.

**Figure 9: Simulated I-V characteristics of HfO**

_{x}memory for different reset stop voltages also demonstrate abrupt set and gradual reset.

Reset pulse transient waveforms are shown for the experimental (see figure 10) and simulated (see figure 11) cases.

**Figure 10: Experimental pulse transient current in the reset process. Current fluctuation is observed, and before the end of pulse, current jumps due to new Vo generation.**

**Figure 11: Simulated pulse transient current in the reset process. Current fluctuation is reproduced.**

For both DC and pulse programming, the reset process is gradual and a significant current fluctuation is observed. By tracking the Vo evolution in the simulation, we found found that a current jump corresponds to a new Vo generation in the gap region at that moment. At the beginning of the reset process with a short gap region, the current fluctuation is caused by the competition between Vo generation process due to the presence of a large field and recombination process due to the presence of mobile oxygen ions migrated from the interface. It is suggested [13] that multilevel HRS states can be achieved (a) by linearly increasing the reset pulse amplitude (see figure 12 and 13) or (b) equivalently by exponentially increasing reset pulse width (see figure 14 and 15).

**Figure 12: Experimental multilevel states achieved by linearly increasing reset pulse amplitude.**

**Figure 13: Simulated multilevel states achieved by linearly increasing reset pulse amplitude.**

**Figure 14: Experimental multilevel states achieved by exponentially increasing reset pulse width.**

**Figure 15: Simulated multilevel states achieved by exponentially increasing reset pulse width.**