Detection of Low-Level Copper-Contamination on Silicon Surfaces by Drop Nucleation
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Fhydrogen-terminated
409
Mat. Res. Soc. Symp. Proc. Vol. 477 01997 Materials Research Society
unit area as (1) J = J 0 exp(-W * /kT). In the case of a non-uniform surface where a fraction f of the surface is covered by a nucleant (contaminant) layer, the form of the expression remains the same. The pre-exponential in the absence of surface diffusion is
Jo = f
(272mkT)12] s n
(2)
where p is the vapor pressure, m is the molecular mass, s* is the area of the critical nucleus, and n is the number of atoms per unit area. For f=l, a typical value for J0 is 1025 cm- 2 sec- 1 . The of the nucleus is critical work for formation 16ira 3
W *=
2
g(0)
(3)
3(p kT In S) where p is the number of atoms per unit volume in the liquid, S is the supersaturation ratio (actual vapor pressure over equilibrium vapor pressure), a is the surface tension between liquid and vapor, and (4) g(O) = (2 + cos 0)(l - cos 0) 2 4 where 0 is the contact angle of the liquid on the contaminant layer (see Figure 1). The radius of the critical nucleus is given by
2a
(5) sin 0. p kT In S The contact angle of the contaminant layer strongly affects the nucleation rate and the radius of the critical nucleus, as shown in Figure 2 for two different supersaturation ratios. Because of the strong dependence, the nucleation rate of drops on a hydrogen-terminated silicon surface (which has a contact angle around 640-78o, see below) contaminated with copper (which has a contact angle near zero [4]) is many orders of magnitude greater than the rate on an uncontaminated hydrogen-terminated surface, even for very low values of f. For contact angles 10 S=.001 15 less than 900, lowering 3" the contact angle leads " 10 to a lowering of both 0o 2F 10 ,2 the critical work for •0 10-0 0 S nucleation and the 101 r_ radius of the critical --------, S =1.001 10 When nucleus. -,-._. 10-45 nucleation occurs at a 101 ' supersaturation ratio of • 10Z 10-60 1.001, the radius of the critical nucleus for 0 40 80 120 160 observable nucleation Contact Angle (degrees) rates (i.e., on the order of 1 cm- 2 sec- 1 ) is about 50 nm as can be Figure 2. Nucleation rates (solid lines) and critical nucleus radii seen in Figure 2. Note (dashed lines) for two supersaturationratios S=I.O01 and S=2 with that increasing the Jo=1O25 cm' 2 sec"1 , T=292.5K, and 3-=76 dyne cm- 1 . r*=
410
supersaturation (S-1) proportionally decreases the critical nucleus size. When hydrogen-terminated silicon is immersed in copper-contaminated ultrapure water, copper is deposited onto the surface through a redox reaction in the form of particles [5, 6]. Because the particles are non-planar, the above theory must be modified to take into account the curvature of the copper deposits. For deposits with radii of curvature much greater than the radius of the critical nucleus, the nucleation rate is not affected by non-planarity. For deposits with radii of curvature on the order of the critical radius, the nucleation rate is lower than in the planar case, but strong preferential nucleation at low supersaturat
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