Electrostatic Theory of Metal Whiskers

Metal whiskers often grow across leads of electric equipment and electronic package causing current leakage or short circuits and raising significant reliability issues. The nature of metal whiskers remains a mystery after several decades of research. Here, the existence of metal whiskers is attributed to the energy gain due to electrostatic polarization of metal filaments in the electric field. The field is induced by surface imperfections: contaminations, oxide states, grain boundaries, etc. A proposed theory provides closed form expressions and quantitative estimates for the whisker nucleation and growth rates, explains the range of whisker parameters and effects of external biasing, and predicts statistical distribution of their lengths.

Figure 1

Scanning electron microscope (SEM) pictures of tin (left) and zinc (right) whiskers. Courtesy of the NASA Electronic Parts and Packaging (NEPP) Program [6].

Figure 2

Sketch of two whiskers of length $h$ and diameter $d$ on a metal surface with local electric fields $\mathbf{E}$ (of opposite directions) inducing the dipole moments $\mathbf{p}$.

Figure 3

Free energy of a whisker vs its length.

Figure 4

The perpendicular electric field component $\mathbf{E}$ at small distance $r$ from the metal surface. Gray circles represent point surface charges with the average intercharge distance $n^{-1/2}>r$

Figure 5

Sketch of the patched area domain of radius $r$ where $+$ and $-$ represent positively and negatively charged patches (shown in dash) of characteristic linear dimension $L$ each. The thick arrow represents the random field vector at distance $r$ along the domain axis.

Figure 6

Sketch of the whisker growth rates vs time. Solid lines represent the two limiting cases given by Eqs. (21) and (22) within the domains of their applicability ($h\ll L$ and $h\gg L$, respectively). The dashed curves are formal solutions of Eqs. (21) and (22) beyond that domains where they must be sewed. The dash-dotted line shows a hypothetical sewing that would fit the observations of Ref. [61].

Figure 7

Sketches of the coordinate dependencies of random electric field $E(h)$ (top), random functional $\xi \equiv {\int }_0^{h}E(x)dx$ and its square (middle), and free energies: surface contribution $F_S$, electrostatic contribution $F_E$, and their sum $F=F_E+F_S$. The vertical dashed line marks the point of $\xi =0$ of $F_E$ flattening that gives rise to a barrier $W_B$ blocking the whisker growth. The barrier $W$ is the same as in Fig. 3.

Figure 8

Left: Probabilistic distributions of Eq. (26) for two different values of the numerical parameter $\gamma$. Right: Comparison between probabilistic distribution of Eq. (26) (open circles) and a log-normal fitting curve (solid line).