Phase-Field-Crystal Model for Electromigration in Metal Interconnects

We propose an atomistic model of electromigration (EM) in metals based on a recently developed phase-field-crystal (PFC) technique. By coupling the PFC model’s atomic density field with an applied electric field through the EM effective charge parameter, EM is successfully captured on diffusive time scales. Our framework reproduces the well-established EM phenomena known as Black’s equation and the Blech effect, and also naturally captures commonly observed phenomena such as void nucleation and migration in bulk crystals. A resistivity dipole field arising from electron scattering on void surfaces is shown to contribute significantly to void migration velocity. With an intrinsic time scale set by atomic diffusion rather than atomic oscillations or hopping events, as in conventional atomistic methods, our theoretical approach makes it possible to investigate EM-induced circuit failure at atomic spatial resolution and experimentally relevant time scales.

Figure 1

Stress evolution in an interconnect: (a) short strip with length $l/a=48$ and (b) long strip with $l/a=384$ for the same time span shown in (a). Strip width is $h/a=8\sqrt{3}$, where $a$ is the interatomic distance of triangular PFC lattice. Applied EM bias $Z^{*}e\mathrm{\Delta }V/(k_{B}T)=0.01$ per unit cell. PFC parameters are $r=0.145$, average density $\overline{n}=0.1$, $P_0$ is the zero current stress, and the noise is set to zero.

Figure 2

Time to reach a critical vacancy concentration as a function of the electric current. Strip length $l/a=48$, height $h/a=8\sqrt{3}$, and $Z^{*}e{j}_0/({\sigma }_0{k}_{B}T)=0.01$. Other PFC parameters are as in Fig. 1. The critical vacancy concentration is assumed when the local pressure reaches $P/P_0=5$.

Figure 3

Nucleation of a void near a triple junction. Atomic density is shown. Panels (a)–(d) are snapshots at times $t=0$, $t=16000$, $t=32000$, and $t=48000$ (in units of $dt$) after the applied electric field is turned on. A strip geometry is used with $l/a=96$, $h/a=32\sqrt{3}$. EM bias $Z^{*}e\mathrm{\Delta }V/(k_{B}T)=0.025$ per unit cell and noise level $2\mathrm{\Gamma }/(\overline{\rho }{a}^{d}dxdy\sqrt{dt})=0.01$. The average density $\overline{n}=0.11$. Other PFC parameters are as in Fig. 1.

Figure 4

Electric potential near a migrating void. The potential fields are overlapped with a coarse-grained PFC density field in (a), (c), and (d): (a) the configuration right after the electric potential is applied; (c) configuration after void migration; (d) with a larger electric field, the void deforms during migration. (b) Deviation of the electric potential ($V_1$) due to the void scattering in configuration (a), with a color bar indicating the relative deviation amplitude $V_1/\mathrm{\Delta }V$, where $\mathrm{\Delta }V$ is the applied potential. Lighter color in the potential plot corresponds to higher voltage. Strip geometry $l/a=96$, $h/a=64\sqrt{3}$. EM bias $Z^{*}e\mathrm{\Delta }V/(k_{B}T)=0.01$ per unit cell in (a)–(c), $Z^{*}e\mathrm{\Delta }V/(k_{B}T)=0.03$ in (d), and noise level $2\mathrm{\Gamma }/(\overline{\rho }{a}^{d}dxdy\sqrt{dt})=0.04$. Basic PFC parameters $\overline{n}=0.1$, $r=0.152$, and others are the same as in Fig. 1.