Intrinsic Stress in Sputtered Thin Films
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- .Fi[rl,r
2 ....
,rN] I
for i = 1,2 ... N,
where m is the mass of the particles, ri is the position coordinates of the ith particle and Fi is the vector force on the ith particle: Fi = dEtot
dri
For pairwise potential, the force Fi on the ith particle is the summation of all pair force Fij between ith and jth particles (j = 1 to N), with the pair force Fij defined as: 131 Mat. Res. Soc. Symp. Proc. Vol. 356 01995 Materials Research Society
Fi
F
d(D(rij) drij
where (D(r) is the interatomic pair potential. For reasonable turn around time, the number of particles N is around 103 and the total simulation time t is around 104 time-steps, with a time-step typically in order of 10-15 second for convergence. With a small number of particles, periodic boundary conditions are normally used to avoid surface effects. SIMULATION MODEL In this study, a deposition system of Mo particles on crystalline Mo substrate was simulated. The Mo substrate composes of 500 particles in 5x5x5 BCC lattice structure. The lattice constant of Mo is 0.43 nm, which yields a Mo substrate dimension of approximately 1.8 x 1.8 x 1.8 nm. Periodic conditions were used in the x and y directions to avoid surface effects. The z direction had a free surface condition, resulting in top and bottom free surfaces. Each Mo particle is introduced sequentially above the top surface. The thickness of the Mo substrate is chosen large enough to avoid system instability due to the surfaces. The piece-9 8 wise pair potential fitted to Mo developed by Miller and rescaled by Ding and Andersen was used. The properties are calculated as followed (the brackets denote time average): Stress tensor:
-i=
'.
m
,
dF
-
V
System stress:
'rii)i 1j
P = Trace(H)
=
1
.(Hj, + [122 + -33)
MEASUREMENTS OF STRESS IN THIN FILMS The stress in thin films was determined by the laser scanning technique, 10 which measures the amount of bending or curvature of the substrate caused by the films. For the case in which the total thickness of the films is much less than that of the substrate, in a substrate that is elastically11 isotropic in the plane of the film, the stress in the film is given by the Stoney equation: 2
E t ESts K, 6(1-,)t where Es and y are the substrate biaxial modulus and Poison ratio, ts and t are the substrate and film thicknesses, and K is the substrate curvature. For a substrate that has an initial curvature before the film is deposited, then the substrate curvature in the equation is modified to AK, the change in the curvature before and after the film is deposited. Stress results in thin films with thickness in the order of a few nanometers however may not be obtained with high accuracy. As the thickness of the films decreases to a few nanometers, the effects of the films on the curvature of the substrate may be minimal that determination of stress is difficult. In addition, the thickness of the contaminant layers or the reaction layers on top of the films may be comparable to the thickness of the films that the measured stress is no longer represe
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