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L. M. KLINGER,t E. E. GLICKMAN,t V. E. FRADKOV,tt W. W. MULLINS# and C. L. BAUER#

Mhe Technion, Haifa, Israel The Hebrew University, Jerusalem, Israel ttRensselaer Polytechnic Institute, Troy, New York, USA I#Carnegie Mellon University, Pittsburgh, Pennsylvania, USA ABSTRACT

The effect of surface and grain-boundary diffusion on interconnect reliability is addressed by extending the theory of thermal grooving to arbitrary grain-boundary flux. For a periodic array of grain boundaries, three regimes are identified: (1) equilibrium, (2) global steady state, and (3) local steady state. These regimes govern the stability of polycrystalline materials subjected to large electric (electromigration) or mechanical (stress voiding) fields, especially in thin films where grain size approximates film thickness. 1. INTRODUCTION Many properties of polycrystalline materials are affected by the intersection of grain boundaries with external surfaces, especially in the presence of applied or internal fields. If the grain boundaries do not transport matter, the corresponding profile evolves via surface diffusion under well known conditions of scale and temperature. 1 If a grain boundary flux I is present, however, the profile evolves by a combination of surface and grain boundary diffusion, which may result in rapidly growing ridges (I > 0) or slits (I < 0). The purpose of this article is to assess the effect of both surface and grain-boundary diffusion on interconnect reliability by extending the theory of thermal grooving to arbitrary grain-boundary flux for a periodic array of grain boundaries. A 2 complete analysis is presented elsewhere. 2. THERMAL GROOVING FOR ARBITRARY GRAIN-BOUNDARY FLUX

Thermal grooving occurs in order to establish local equilibrium at the intersection of a grain boundary with an external surface. In the symmetric case of a boundary normal to an originally flat surface (x = 0), the angle of inclination of the positive branch of the surface at the groove root with respect to the x axis is (1)

0 = sin-1 I,

where 1i - Ygb/ 2 ys and Ygb and Ys denote, respectively, grain-boundary and surface free energy.

Under conditions of mass continuity at the groove root and isotropic surface energy, the corresponding profile y(x) is governed by the expression1 y -(Js)x =-B{ (1 + y2)-1/ 2 [(1 + y2)-3/ 2 Yxx]x} x (2)

where subscripts denote differentiation and B = 8SDSQys/kT in which 6,, D,, Q, k, and T denote,

respectively, thickness of the surface diffusion layer, surface diffusion coefficient, atomic volume, 295

Mat. Res. Soc. Symp. Proc. Vol. 391 ©1995 Materials Research Society

Boltzmann's constant, and absolute temperature. If y 0 are functions of IL.When ca < 0 (matter is removed from the surface) the groove becomes deeper, whereas, when c > 0 (matter is supplied to the surface) the groove becomes shal-

lower. In all cases, the surface assumes a steady-state profile which translates at velocity V. = vS1/vo = ca,where v. = B/L 3. A typical steady-state profile is illustrated schematically in Fig. 2. When ca f