Advances in Condensed Matter Physics

Volume 2015, Article ID 136938, 8 pages

http://dx.doi.org/10.1155/2015/136938

## Model of Reversible Breakdown in HfO_{2} Based on Fractal Patterns

Department of Information Engineering, Electronics and Communications, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy

Received 21 October 2014; Revised 27 December 2014; Accepted 4 January 2015

Academic Editor: Jan A. Jung

Copyright © 2015 P. Lorenzi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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-*k* based metal-insulator-metal (MIM) capacitors is today strongly desired for application in resistive switching memory (RRAM). In fact, among the technology options for nonvolatile memory devices, RRAM is gaining an increasing consideration for its superior scaling opportunities, high speed, and low operating voltages [1–3]. For all these reasons, understanding reversible breakdown in high-*k* dielectric films is today of major interest for microelectronics. The metastable soft breakdown of the dielectric film is due to the built-up and rupture of conductive filaments (CF) formed of defects which can be achieved by consecutively applying proper voltages at the two electrodes. While the key factors driving the resistive switching have been identified and already widely modeled [4–11], the kinetics of CF formation is still under debate. At an atomistic scale, some insight in the time evolution of filaments is given by Monte Carlo modeling in 2D [12] and 3D [13], which relates the time to breakdown to the applied voltage and the filament evolution to vacancy migration. The model proposed here describes the kinetics of growth of conductive filaments until breakdown within an external driving factor (voltage pulses ) in terms of fractal aggregation of defects.

#### 2. Method

The damage structures observed in dielectric breakdown in many cases have the form of trees [14–18]. A tree is at least at some level of approximation a fractal [15] or self-similar object in the sense that a branch of the tree again looks like a tree and so on. In fractal structures the relation of the total length of all branches inside a circle of radius (the number of all the grid points) and the radius itself is a power law with noninteger exponent FD: , where FD is the fractal dimension. 2D numerical modelling of dielectric breakdown in terms of fractal patterns of defects was already proposed in the past, in the case of solid [19], liquid [20], and gaseous [21] materials. In all those simulations and visual inspections, breakdown in dielectric media was always found to feature fractal patterns with . In the present case the dielectric breakdown in metal-insulator-metal (MIM) capacitors is studied. The defect distribution is filamentary, growing from one electrode to the other. The condition will be respected. The filaments are assumed to be electrically conducting so that the field at the filament tip exceeds the original field at the electrode plate. This leads to an amplification, propagation, and multiplication (branching) of the local field enhancement. So, once a characteristic value of the electric field is exceeded, a rapid and abrupt increase of the charge density near the filament tip is achieved. The avalanche behaviour is a key feature of the phenomenon descending from its fractal or self-similar nature. With respect to previous study of breakdown of dielectrics in MOS capacitors, in the present model there are two fundamental improvements: (i) the probability that a defect is (or is not) generated depends on the maximum value of electric field in the mesh and (ii) the position in the grid where the defect is generated depends on the local electric field. The cross-section of the oxide film is meshed by a 2D grid. The matrix is given by elemental cells with size (representing a point). At a generic time , a distribution of defects is present, as depicted in Figure 1 (with , ): black cells of the matrix represent point defects. Breakdown occurs when defects are piled up in a way to connect the two electrodes. The distance to be covered to complete a filament is (dashed arrows in Figure 1) and the minimum distance in the whole matrix is (bold red arrow). In many situations the local electric field in each row of the matrix is higher than due to the needle shape of neighbor filaments and this accelerates the filament build-up. A rigorous evaluation of the field distribution would require burdensome numerical simulations. In our treatment, it will be posed and the slowest case of breakdown will be considered. This leads to possible underestimation of the times to breakdown. The steps of the modelling procedure are resumed in the flow diagram shown in Figure 2.