Corresponding author: Andrey A. Orlov (aorlov@niime.ru)

In this work, the simulation of the processes of diffusion of metal barrier ions into a low-k dielectric between two nearby copper lines was performed. Based on experimental data on the diffusion coefficient published in the scientific literature and calculations according to the mathematical model of the distribution of metal barrier ions in the dielectric, the time dependent breakdown of a porous low-k dielectric in the elements of very large-scale integrated circuits of the modern topological level was estimated. Additionally, the work obtained dependences of the dielectric breakdown time on the distance between two nearby copper lines along with dependence on the power voltage of the line (the other line is grounded).

With the permanent reduction of topological level in the fabrication of very large scale integrated circuits

The main causes of VLSI degradation in copper metallization systems for ≤90 nm topological level are copper ion diffusion and drift into the dielectric under an external electric field [

The cause of TDDB is believed to be the formation of a conducting layer of traps connecting two nearby metallization lines (electrodes) which eventually leads to a significant current increase. It is believed that the traps (localization centers of electrons tunneling from electrodes) are formed by diffusion of metallic barrier ions (e.g. Ta/TaN) [

where _{i}_{,j} is the distance between the _{i}_{,j} is the energy barrier between the two centers, _{B} is Boltzmann’s constant and

The traps form a network of resistors where the resistivity _{i}_{,j} between the _{i}_{,j}. Furthermore, the distance _{i}_{,j} in the 2D system is determined by the concentration of the metallic barrier ions _{i}_{,j} = ^{–1/2} [

Schematic of physical processes in simulated structure containing two nearby copper lines separated by dielectric [

The aim of this work is to develop a mathematical model of the distribution of metal barrier ions in a low-k dielectric and to estimate the influence of the input parameters of the model on the numerical value of the time dependent breakdown of the dielectric.

The normalized minimum concentration of the metal ions ￼ can be determined from the equation of diffusion and ion drift in electric field (Eq. (2)) and the boundary conditions at the electrodes (Eq. 3)) [

_{norm} (_{norm} (

The calculations of the normalized metallic ion concentration between two nearby copper lines as a function of time as illustrated in Fig. ^{8} s (≈3.96 years). Similar calculations were also carried out for earlier reported parameters [

Distribution of ion concentration in metallic barrier between two nearby copper lines. Input parameters: _{0} exp(-_{a}/_{B}_{0} = 2 · 10^{-11} m^{2}/s, activation energy _{a} = 0.9 eV and

Determination of time to failure (TTF) of porous dielectric

Distribution of ion concentration in metallic barrier between two nearby copper lines. Input parameters: ^{-20} m^{2}/s;

The required input parameters for simulation can be obtained from experimental data. It was shown [^{1/2}, where _{0} and the diffusion activation energy _{a} can be derived from Eq. (4) using the following well-known relationship between the diffusion coefficient and temperature:

_{0} exp(-_{a}/_{B}

Diffusion length can be measured, for example, using transmission microscopy. Also, Arrhenius’ graph can be calculated from X-ray diffraction profiles [

Figure

For the parameters used for the simulation in Fig. _{1} = 3.33739, _{2} = 3.33739, _{1} = 55.8961, _{2} = 68.31746, and TTF_{0} = –8.66352.

In this work the potential difference (or, equivalently, the electric field) between two nearby lines is accepted to be constant which is the case for the following bias line powering schemes: powered line / grounded line, powered line / periodic signal, periodic signal / periodic signal. The case “powered line / periodic signal” will be analyzed in further studies.

Figure _{DD} (the other line being grounded) at constant diffusion coefficient

Approximation of the resultant function also has an exponential pattern and is described by the following expression:

For the parameters used for the simulation in Fig. _{1} = 67.11307, _{1} = –0.24943, and TTF_{0} = 3.17971.

TTF as a function of distance between copper lines (at constant

TTF as a function of power voltage _{DD} (the other line being grounded) at constant

Thus this work reports a method for estimating the time to failure of a porous dielectric in VLSI systems that is based on the determination of the diffusion coefficient from experimental data and calculations within a mathematical model of ion distribution for a metallic barrier in the low-k dielectric between two nearby copper lines. Additionally the TTF was calculated as a function of the distance between two nearby copper lines and as a function of one line’s power voltage (the other line is grounded). These functions are showed to be exponential. Further studies are planned for analyzing the change in the TTF with a periodic signal being supplied to one line (the other line is grounded) as well as the TTF as a function of integrated circuit topology geometry.