February 2015 • SMT Magazine 43
Patch structured electric fields near metal
surfaces have been found in multiple works. The
measurements reveal the work function fluctua-
tions of ~0.5 eV induced by L~10 μm patches in
some metals.
The charged patch model enables one to
estimate the average electric field vs. distance r
from the surface. At distances r<>L, fluctuations become essential and the av-
erage field amplitude decays inversely propor-
tional to r. In addition, given the above model,
it is straightforward to see that far from the sur-
face, r>>L, the electric field vectors are directed
on average perpendicular to the surface having
a considerable dispersion in angles,
equation 1
where E
^
and E
||
are respectively the field
components perpendicular and parallel to the
surface, and angular brackets represent averag-
ing.
9.
The above estimates lead to the follow-
ing scenario of whisker evolution. (i) Stage 1:
Whiskers nucleate in time intervals of one sub-
second to one day (reflecting fluctuations in
nucleation barriers due to the local field fluc-
tuations); their dimensions upon nucleation
are h~10–100 nm and d~1–10 nm, with the
average orientation perpendicular to the metal
surface and significant angular dispersion. (ii)
Stage 2: Whiskers grow up to the patch size,
say L~0.1–10 μm. This takes a much longer
time t
0
~10
4
–10
5
that can be experimentally
identified as the whisker incubation time. The
growth rate at this stage is very low for almost
entire time interval t
0
, with drastic acceleration
in the nearest proximity of t
0
(see Figure 6).
(iii) Stage 3: Whiskers grow way above patch
size at the average constant rate possibly with
some degree of winding or kinking (beyond
the current theory). At this stage, random field
configurations induced by uncorrelated patch
charges make growth rates of individual whis
-
kers fluctuating, some of them blocked. The
random distribution of blocking barriers de-
termines the statistical distribution of whisker
lengths. (iv) Stage 4: If whiskers grow above
lengths where feeding by thermal radiation
dominates, they evolve further in lateral di-
rections parallel to the metal surface (not dis-
cussed here).
Figure 15: Charged patch model. The correlation function g(x) can be approximated as the delta-
function (i.e., uncorrelated disorder) at distances x>>l.
eLeCTrOSTaTIC MeCHaNISM OF NuCLeaTION aNd GrOWTH OF MeTaL WHISKerS continues
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