Electromigration of Intergranular Voids in Metal Films for Microelectronic Interconnects

Voids and cracks often occur in the interconnect lines of microelec tronic devices They increase the resistance of the circuits and may even lead to a fatal failure Voids may occur inside a single grain but often they appear on the boundary between two grains In this work we model and analyze numerically the migration and evolution of an intergranular void subjected to surface di usion forces and ex ternal voltage applied to the interconnect The grain void interface is considered one dimensional and the physical formulation of the elec tromigration and di usion model results in two coupled fourth order one dimensional time dependent PDEs The boundary conditions are speci ed at the triple points which are common to both neighbor ing grains and the void The solution of these equations uses a nite di erence scheme in space and a Runge Kutta integration scheme in time and is also coupled to the solution of a static Laplace equa tion describing the voltage distribution throughout the grain Since the voltage distribution is required only along the interface line the two dimensional discretization of the grain interior is not needed and the static problem is solved by the boundary element method at each time step The motion of the intergranular void was studied for dif ferent ratios between the di usion and the electric eld forces and for di erent initial con gurations of the void Introduction Reliability of microelectronic circuits depends to a large extent on prop agation of damages in the interconnect lines Voids and cracks increase the resistance of the line and may even lead to open circuits thus dramatically decreasing the lifetime of the device Failures occur trough the void evolution growth and motion which are driven primarily by surface di usion electric eld and mechanical stresses Studies on grain void interface dynamics and void morphology make it possible to predict the interconnect performance and eventually to improve the quality of a circuit design Voids often occur in grain boundaries and thus studying and modeling their surface di usion and electro migration phenomena requires some in tergranular analysis In this paper we develop a numerical approach to track the dynamics of the void motion under the action of di usion forces and an external voltage Mechnical stresses are not treated in this paper Consider a curved line which separates the grain from the void The basic assumption is that the interface moves in its normal direction with a known velocity function F The tangential motion is usually neglected when a void inside a single grain is considered For an intergranular void the tangential motion is allowed along with the normal motion in the proximity of triple points which belong to each of the two grains and to the void Otherwise the boundary conditions can not be satis ed The front velocity is caused by the surface di usion and the electromigration components depending on the second derivatives of the interface curvature K and of the voltage U respectively The derivatives are taken with respect to the interface arclength s F B Kss Uss The positive coe cients B and express the contributions of the di usion forces and the eld forces respectively They depend on the properties of the material temperature etc and are considered here to be material constants We assume further that the time is normalized so that B The structure of the paper is as follows in section we review some of the central works on electromigration in thin metallic lms In section we derive the physical formulation of the two grain metallic lm problem We consider the motion of triple points and the continuity conditions at the triple joints We distinguish between a general case and a symmetric void where the axis of symmetry coincides with the grain boundary line and with the direction of the external electric eld In section we give a nite di erence formulation for the di usion component of the normal velocity of the interface We consider rst a general internal point of the interface and then speci cally a triple point referring both to the normal and the tangential motion at the proximity of the triple point In section we consider the elliptic equation for distribution of the electric potential and formulate its boundary conditions The numerical results are presented in section Electromigration in thin lms Review Predictions for dyanmics and evolution of voids and cracks in thin metallic lms of interconnect lines have a practical importance for microelectronics techology and many researchers have contributed to this study In this sec tion we review what we consider as the most essential Xia et al intro duced a two dimensional nite element method to compute the motion and evolution of a void in a single grain by surface di usion in an elastic elec trically conductive solid They accounted for the in uence of deformation electric current ow and e ects of mass ow The methodology has been used to predict the conditions of failure of the interconnect A three dimen sional nite element model for studying the motion and evolution of voids in an idealized interconnect line due to electromigration and strain induced surface di usion have been developed by Zhang et al to predict more ac curately the behavior of the void The spatial nite element formulation has been further exploited by Zhang and Bower to study the conditions for formation of islands in a thin lm of semiconductor material Kraft tet al presented the results of experimental studies of the mechanical behavior of thin lms Films on the silicon and polymer substrates were strained and the lm stress was determined by X ray measurements Kraft and Arzt examined electromigration mechanisms in conductor lines applying nite di erence and nite element formulations and com pared the results of numerical simulations with experimental studies They analyzed an electromigration damage in unpassivated aluminum based in terconnect lines and discovered that slit like voids in bamboo lines develop from equi axed voids through a shape change driven by the electron wind taking into account the nite width of the line surface tension e ects and anisotropic di usivity Further Kraft and Arzt developed a model to predict lifetimes of conductor lines and to describe the microscopic damage behavior considering nucleation and growth of electromigration voids The reliability of a bamboo line is governed by the critical current density Experimental and theoretical studies of aluminum interconnects that in clude copper as an alloying element were carried out by Spolenak et al A mechanistic model was developed to account for an increase in activation energies for aluminum drift in alloys in comparison to the pure metal and this approach was in good agreement with the experiments Copper as an alloying element improves strongly the electromigratrion resistance of the in terconnects and thus reduces the damage caused by evolution and growth of voids Gungor and Maroudas in a series of papers and with collaborators studied the complex of linear and non linear dynamics and stability of transgranular voids in thin lms with bamboo grain struc ture They simulated the formation of various morphological features void faceting formation of wedge shaped void propagation of slit like and soliton like features and open circuit failures In they performed a the oretical analysis for conditions of failure of metallic lms The dynamics of void is governed by a surface mass transport electric eld elastic and plas tic deformations Anisotropic properties are taken into account in surface di usivity and stresses In they applied a linear stability theory to ex amine the e ects of anisotropy in surface di usion In they performed a theoretical analysis and numerical simulations of the e ects of residual stress on the failure mechanisms of passivated metallic thin lms and observed two modes of failure to occur concurrently during void morphological evolu tion electromigration induced formation of faceted slits and stress induced formation of ne scale crack like features on the void surface In current induced morphological evolutions of void surfaces were studied theoretically and numerically accounting for surface di usional anisotropy were studied The non linear analysis predicted a surface morphological transition and the onset of oscillatory dynamics at a critical strength of the applied electric eld Migration of voids along the lm at constant speed with steady and time periodic surface morphologies was studied for electric elds weaker and stronger than critical In a recent study a multiscale modeling approach for the analysis of morphological evolution and electromigration failure in ductile metallic thin lms of integrated circuits was presented The e ects of anisotropy of void surface di usivity on on the stability of the interconnects was presented in They predicted formation of stable faceted voids and wedge shaped voids and studied failure due to propagation of slit like fea tures from void surfaces In they considered morphological instabilities caused by simultaneous action of applied mechanical stress and electric eld on transgranular dynamics of voids and studied propagation of slits They predicted by numerical simulations that a failure might occur by the coupling of two modes of surface morphological instability The above serious exten sive broad analysis was very instrumental for the design of integrated circuits by allowing to predict the reliability and the lifetime of the interconnect Schimschak and Krug proposed a continuum description of the sur face evolution that takes into account electromigration and capillarity driven surface di usion They applied for numerical simulations a one dimensional model of interface dynamics which can be parameterized by a height func tion The one dimensional geometry is convenient and relevant to the model ing of shape changes at the edge of an e ectively two dimensional conductor line but becomes inappropriate if the dynamics creates overhangs In they studied numerically the motion and shape evolution of an in nitely extended isotropic and homogeneous two dimensional current carrying con ductor considering a series of morphological transitions and conditions of instability for initially circular voids The electromigration induced shape evolution of cylindrical voids was numerically examined in accounting for mass transport current crowding and crystal anisotropy in the surface mobility The authors considered nite strips with periodic boundary condi tions in the current direction and voids in in nite or semi in nite strips Mahadevan et al studied an edge instability in single crystal metal lines applying a numerical phase eld technique The authors de ned the critical value of the applied current when the edge perturbation grows to become a slit shaped void that spans the wire and leads to electrical failure reducing the circuit lifetime to an unacceptable level The phase eld model provides an attractive alternative to these techniques in which the in terface is not explicitly tracked In the phase eld approach the idea of a sharp interface between metal and void is abandoned Instead the thin lm and the space around it are described by a continuous and but rapidly vary ing scalar function called an order parameter It accepts speci c distinct uniform values well within the metal and well inside the void There is an apparent analogy between the level set method and phase eld method In the level set the level function is usually a signed distance from the given point to the interface line The distance is considered negative in the void and positive in the conducting medium Applying the hyperbolic tangent to the scaled distance function one gets a parameter of phase eld Indeed this leads to unity well inside a metal and minus unity well in a void The scaling parameter adjusts the rapidly or slowly varying function Mahadevan and Bradley used a phase eld method to simulate the time evolution of a perturbation to the edge of current carrying single crystal metal line accounting for electromigration surface di usion and current crowding The authors provided a fabrication criterion which ensured that the wire will not fail through formation of voids In this case the edge per turbations are small enough so that the operating current does not exceed the threshold and the instability through the formation slit shaped voids does not occur An alternative approach to increase the lifetime of the in terconnect is to orient appropriately the crystal with respect to the applied eld The phase eld principle with an order parameter characterizing the dam aged state of the interconnect was further used by Bhate et al for simulating electromigration and stress induced void evolution The evolution of the order parameter was governed by a fourth order parabolic PDE re lated to the Cahn Hilliard equation Bhate et al solved the PDE by an implicit nite element scheme together with the accompanying mechanical elastic and electrical problems Fridline and Bower studied the e ect of the anisotropy of the surface di usivity on the formation and growth of slit like voids considering an in terconnect as a two dimensional electrically conducting strip that contains an initial semi circular void They applied a nite element model to predict the evolution of the void after an electric eld was applied to the strip Their later work accounts for several kinetic processes involved in interconnect failures including surface di usion interface and grain boundary di usion sliding on grain boundaries and at the interface between line and elastic passivation Gray et al applied an approximate Green s function to solve the Laplace equation for the electric potential by boundary element method in order to model the void dynamics under electromigration conditions in metallic thin lm interconnects used in integrated circuits The approximate Green s function was constructed by re ecting the source point through a plane The method retains the primary attribute of an exact Green s func tion elimination of surfaces from the calculation Another application of the method is the simulation of the industrial electroforming process Suo considered aluminum interconnects in the presence of insulators and shunts subjected to temperature change and a direct electric current He studied the evolution of interconnects into a stable state with a segment of aluminum depleted near the cathode a linear distribution of pressure in the rest of the line and no further mass di usion and estimated the time scale for the interconnect to evolve to the stable state In the mecha nisms for di usive processes in solid structures of small feature sizes between a few to hundreds of nanometers was described Considering microelectronic and fotonic devices he applied the concept of free energy The change of free energy de nes a thermodynamic force which in its turn drives the con gurational change of the structure The author gives a physical description of forces of diverse origins that occur in thin lms of interconnect lines in cluding elasticity electrostatics capillarity electric current and others Yu and Suo derived a nite element formulation to model the dynamics of a single pore on a moving grain boundary assuming that surface di usion is the dominant process for a small pore to adjust its shape and position Sun et al considered the evolution of grains in a polycrystalline ber and applied a variational approach to microstructure development which in corporates thermodynamic forces and mass transport mechanisms The free energy includes interfacial elastic electrostatic and chemical components The rate process includes di usion creep grain boundary motion and sur face or interface reactions In a later work Sun et al modeled the dynamics of a two grain thin lm on a substrate They developed a nite el ement formulation that accounted for surface tension anisotropy bulk phase free energy density and nite junction mobility The authors formulated the laws for the motion and equilibrium of triple junctions and considered the motion of grain void interfaces and grain boundary The large shape changes of a solid due to matter di usion on its surface were studied by Sun and Suo In addition to surface di usion evaporation and condensation were accounted for and the nite element approach was applied to analyze the thermal grooving on a polycrystalline surface An electromigration problem with a speci c geometry is solved in our study by a di erent numerical approach We applied a parametric presen tation of the curved interface between external void and the bulk material in the Cartesian space In addition the application of the cubic spline tech nique to rearrange the generating points of the interface line in a uniform manner allows to apply the standard methods of nite di erence of any or der We assume that the boundary between the two grains was parallel to the direction of the external electric eld and that the void was symmetric relative to this boundary With this approach one may omit continuous or systematic application of the remeshing and elimination procedure on the irregularly disposed nodes Physical formulation Angles at triple points Consider a grain boundary that is split by the void into two sections as shown in Fig A The vector E denotes the direction of the electric eld The void migrates and evolves under surface di usion and electric eld forces We assume for simplicity that the grain boundaries A A A and B B B contribute no additional electric resistance and suppose that the speci c resistance of both grains is the same We assume that mechanical equilibrium holds at the triple points A and B The speci c surface energy per unit length of of the grain void boundary is the same for grains and Therefore at the triple points the angles between the tangent line to the intergranular boundary and the tangent lines to the void interface contour are the same C AA C AA D AA D AA The value of these angles is de ned by