_{2}Based on Fractal Patterns

We propose a model of the kinetics of reversible breakdown in metal-insulator-metal structures with afnia based on the growth of fractal patterns of defects when the insulator is subject to an external voltage. The probability that a defect is (or is not) generated and the position where it is generated depend on the electric field distribution. The new defect moves accordingly to fractal rules and attach to another defect in a tree branch. When the two electrodes sandwiching the insulating film are connected a conductive filament is formed and the breakdown takes place. The model is calibrated with experiments inducing metastable soft breakdown events in Pt/HfO_{2}/Pt capacitors.

The loss of insulating properties of thin oxide films is due to the phenomenon of dielectric breakdown. The irreversible breakdown is one of major sources of failure in integrated circuits based on metal oxide semiconductor field effect transistors (MOSFET) technology. Reversible soft breakdown of gate dielectrics is not fatal with respect to the MOSFET switching but leads to undesired current leakage through the gate which affects the power performance. On the contrary, deep control of the reversible changes of conductivity between high and low resistance states in high-

The damage structures observed in dielectric breakdown in many cases have the form of trees [

Sketch of the defect distribution in the oxide at two different iteration steps, represented by black cells in the

Flow diagram of the simulation procedure described in the text.

At time zero

Weibull distributions calculated with the proposed model at

The electric field distribution in the matrix is calculated at the step 2 of Figure _{2} films [

In the case of a structure with parallel plane faces, the fractal tree starts from one of the two interfaces, and then the DLA algorithm cannot be used as it is. We modified it in order to generate fractals structures describing breakdown in metal-oxide-semiconductor or metal-insulator-metal structures. In contrast with the DLA, in the present case particle displacements are not random, but probabilities are ruled by quantities

The proposed methodology can be of interest for the study of the electroforming in the resistive RAMs (RRAMs) based on MIM stacks. Electroforming consists in the operation of soft breakdown in fresh devices due to defects (oxygen vacancies) piling up between the two electrodes when the insulator is subject to electrical stress. In the last years, the phenomenon of electroforming in RRAMs has been widely studied experimentally and simulated (see, e.g., [

The model has been calibrated using experimental results obtained on MIM capacitors of the type Pt/HfO_{2}/Pt. The HfO_{2} film (10 nm thick) was ALD deposited on sputtered bottom electrode. Also the top electrode was sputtered and the stack patterned via hard mask etching. Other details on sample preparation, material analysis, and complete characterization of samples in quasi-static condition are reported elsewhere [

Comparison between experiments performed on RRAM cells of the type Pt/HfO_{2}/Pt. HfO_{2} and simulations: (a) measured (symbols) and calculated (lines) values of

It is worth noticing that the voltage behavior of _{2}). Once the calculated trend reproduces the experimental one, there is another step regarding times to be completed. In fact, the output of our modeling procedure does not contain any information about absolute times, since the discretization is given by iterations. Therefore, in order to interpolate measured data, a trivial normalization is made, forcing the calculated point at

Recalling the thermochemical model proposed to explain the field-dependent breakdown in thin SiO_{2} films [^{−1}, which means that with activation energies of a few eVs the local temperature for defect generation is several hundreds of kelvin degrees.

The simulation procedure yields the defect distribution at each iteration. Its evolution until soft breakdown is shown in the 3D plot shown in Figure

Time evolution of 2D distribution of defects calculated with the proposed model in the specific case of

Further considerations may be done about the rate of generated defects. The (minimum) number of defects (

(a) Minimum number of generated defects as function of time, for different operation voltages. (b) Calculated rates (symbols) and simulated derivatives (lines) of the curves drawn in (a).

Those curves may be derived numerically and the calculated rates ^{8} defects/s, which means that generation of one defect takes approximately 1.6 ns at the maximum electric field. This value should be interpreted as an indication of the timescale of the phenomenon, since it is affected by the 2D approximation. Finally, the voltage and thickness dependence of the number of defects at the moment of soft breakdown (

Maximum electric field calculated using the fractal growth of the conductive filament for several operation voltages (symbols) and interpolating analytical functions given by the sum of two exponentials (continuous lines).

Voltage dependence of the average number of defects at the moment of soft breakdown, drawn for different thicknesses of the oxide film.

It is proposed a model of the formation kinetics of conductive filaments in oxides subject to an external driving factor (voltage pulses). The model is based on fractal aggregation of defects and is inspired by literature works extensively reporting the fractal nature of breakdown events in solids, liquid, and gases, where a fractal dimension around 1.8 was reported. It proposes a modified version of the diffusion limited aggregation model of breakdown fractal trees, in which defect displacements are not random, but probabilities are ruled by quantities which definitely fix to 1.8 the fractal dimension of the tree. It takes into account the presence of native defects and a defect generation probability depending on the local electric field. With respect to other models, the proposed approach predicts the time rate of defect formation, the avalanche onset of breakdown, and the dependence of the average number of defects at breakdown on the insulator thickness and the applied voltage. The proposed methodology can be of interest for the study of the electroforming in the resistive RAMs (RRAMs) based on MIM stacks, which are of outmost interest in microelectronics being candidate to replace floating-gate nonvolatile memories in the future nodes. The model is calibrated comparing simulation results with experiments performed on Pt/HfO_{2}/Pt MIM cells designed for RRAM applications. Excellent simulations of the time to soft breakdown versus voltage curves and of the Weibull distributions are obtained meshing the structure. A 2D approximation has been assumed for brevity, since it allows getting unexplored insights on the breakdown phenomenon with reasonable simulation times. The study of the time to breakdown reveals a sort of avalanche mechanism, with a sudden onset and a rapid conclusion, as predicted in a fractal approach. The correspondence between the physically based model of thermochemical breakdown in dielectrics and the present one yields a proportionality between the local temperature where the defect is created and the activation energy for the creation of the new defect. The average number of defects building-up in a filament can be predicted as function of time and is not dependent on voltage. The timescale for generation of one defect is estimated in the order of a nanosecond. Similarly, the number of defects to breakdown can be predicted as function of voltage and film thickness. As a result, the number of defects in a conductive filament does not depend on voltage and increases linearly with the film thickness.

The authors declare that there is no conflict of interests regarding the publication of this paper.

_{2}based RRAM

_{2}-based RRAM cells

_{2}RRAM devices

_{2}for statistical reliability prediction

_{2}-based RRAM

_{2}: a kinetic Monte Carlo study

_{2}films with fractal distribution of traps

_{2}films and of the interface with TaN electrode

_{2}O

_{3}interface

_{2}dielectrics