232 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 2, FEBRUARY 2002 A Study of Soft and Hard Breakdown—Part I: Analysis of Statistical Percolation Conductance Muhammad Ashraful Alam, Senior Member, IEEE, Bonnie E. Weir, Member, IEEE, and Paul J. Silverman Abstract—A theory of the statistical origin of soft and hard breakdown, that can explain a wide range of experimental data, is proposed. The theory is based on the simple premise that the severity of breakdown depends on the magnitude of the power dissipation through the sample-specific, statistically distributed percolation conductance, rather than on any physical difference between the traps involved. This model (a) establishes the con- nection between the statistical distribution of the theoretically predicted percolation conductance and the distribution of ex- perimentally measured conductances after soft breakdown (Part I), and (b) explains the thickness, voltage, stress, and circuit configuration dependence of soft and hard breakdown (Part II). Connections to previous theories are made explicit, and contradictions to alternate models are resolved. Index Terms—Hard breakdown, MOS devices, reliability, semi- conductor device modeling, soft breakdown. I. INTRODUCTION W HEN stressed by an applied voltage, an oxide film loses its insulating properties in two stages [1]–[4]. First, in the wearout phase, traps are generated within the oxide that in- crease the leakage current through the film. Eventually, these traps complete a percolation path that bridges the two elec- trodes across the oxide. Various models, e.g., anode hole injec- tion (AHI) theory [5]–[7], hydrogen release model [8], [9], [35], and electrochemical model [10], etc., have been proposed to ex- plain this wearout phase of oxide degradation. Then, once the percolation path is completed, the power dissipation through it controls the second stage of the breakdown transient (charac- terized by submicrosecond time-constants, segment I-II or seg- ment I-III, see Fig. 1) which determines the postbreakdown con- duction property of the oxide. This second stage of oxide degradation for thinner oxides tested at lower voltages (soft) appears to be fundamentally dif- ferent from breakdown for thicker oxides tested at higher volt- ages (hard) [2], [12]–[17]. Hard breakdown is characterized by a large change in voltage or current during the breakdown tran- sient and a postbreakdown current–voltage ( ) character- istic that is essentially ohmic (see Fig. 1, long dashed line). Soft breakdown is detected by a much smaller change of voltage or current after breakdown and by postbreakdown charac- teristics which can be described by a power law (see Fig. 1, short-dashed line). Although an impressive set of Manuscript received March 20, 2001; revised September 17, 2001. The re- view of this paper was arranged by Editor G. Groeseneken. The authors are with Agere Systems, Murray Hill, NJ 07974 (e-mail: alam@agere.com). Publisher Item Identifier S 0018-9383(02)00821-3. Fig. 1. One monitors the pre- and postbreakdown characteristics to determine soft (nonlinear, power-law conduction: short dashed line) versus hard (ohmic conduction: long dashed line) breakdown. data has been collected for soft-breakdown over the years [11], [16], [20], and the possibility of using soft-broken oxides in cir- cuits has been discussed [11]–[13], a consistent theory of soft and hard breakdown that explains available experimental results is still lacking. To be useful for reliability projections and fu- ture oxide scaling, we must be able to translate the available data—collected from large area capacitors stressed at high volt- ages—to ultrasmall IC transistors operating at low voltages for oxides which are yet to be fabricated. Moreover, this scaling must be performed with an eye to the actual circuit conditions which may stress these oxides in some combination of constant voltage and constant current stress. The theory proposed in this paper is an effort to achieve these objectives. In this paper, after the introduction, we propose a theory of breakdown transient (segment I-II or I-III, see Fig. 1) in Sec- tion II. This theory is based on two simple concepts: statistical distribution of the percolation conductance and the exis- tence of a threshold power density . Section III discusses the concept of percolation conductance and its time evolu- tion. Our conclusions regarding the statistical properties of are summarized in Section IV. The companion paper [34] will then analyze the consequences of the statistically-distributed percolation conductance and critical threshold power den- sity in determining hard and soft breakdown as a function of capacitor area, oxide thickness, and stress conditions, estab- lishing the scaling principles for reliability projections. II. CIRCUIT MODEL FOR BREAKDOWN TRANSIENT Consider an oxide capacitor stressed with a constant current stress (CCS) [or a constant voltage stress (CVS)], as shown in Fig. 2. During the wearout phase, the current-flow from the electrodes is balanced by spatially uniform tunneling current through the oxide [which may include both direct or Fowler-Nordheim tunneling current as well as the stress-in- duced leakage current (SILC)]. Therefore, at time (the 0018–9383/02$17.00 © 2002 IEEE