Advancement of a fine slit along a planar grain boundary in an electric fieldE0, applied parallel to the slit, is investigated by considering electromigration along both the grain boundary and the slit surface. Electrically induced flux in the grain boundaryIgb(+ toward the slit tip) and both electrically and curvature‐induced fluxes on the slit surfaces are considered assuming 2Is>Igb, whereIsis the flux (+ away from the slit tip) on each of the parallel slit surfaces far removed from the tip. Steady‐state solutions of the transport equations are classified according to the value of a parameter &bgr;=tan−1(2Is/Igb) which, under reasonable assumptions, depends on material parameters only. For 5&pgr;/4≥&bgr;≥&bgr;2, unique steady‐state solutions exist; for &bgr;2>&bgr;>&bgr;1, multiple steady‐state solutions occur; below &bgr;1≥&pgr;/4, no steady‐state solution is possible. Since &bgr;1<&pgr;/2,Igb>0 (flux exiting the grain boundary into the slit) for all cases in which no steady‐state solution is possible. In the case of multiple solutions, those corresponding to smallest width (and hence largest velocity) are determined. For all steady‐state solutions, slit width and tip velocity scale asE−1/20andE3/20, respectively. Results also apply to the propagation of a slit within a grain or along a passivation layer. Generally, tip velocities can approach 1 nm/s (3.6 &mgr;m/h), thereby representing a likely failure mechanism in fine‐line (near bamboo structure) interconnects. ©1996 American Institute of Physics.