multi-physics computer simulation ofthe electromigration...

Multi-Physics Computer Simulation of the Electromigration Phenomenon

Xiaoxin Zhu I*, Hiren Kotadia3, Sha xu2

, Hua LUI, Samjid H. Mannarr', YC Chan", Chris Bailey'1. School of Computing and Mathematical Sciences, University of Greenwich,

30 Park Row, London SEI0 9LS, UK2. Department of Electronic Engineering, City University of Hong Kong,

83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China3·Physics Department, School of Natural & Mathematical Sciences, King's College London,

Strand, London WC2R 2LS, UK*[email protected] (+44) 7901-153453

However, in other experiments that followed Black'swork, it was found that not all experimental results can bedescribed by Eqn. 1, and Eqn. 1 was modified to become Eqn.2 where n has been found to vary depending on many factors.

AbstractPrevious works on electromigration in microelectronics

devices have been reviewed, and a multi-physics EMsimulation method that combines electric, thermal, atomicdiffusion, and stress analysis has been described. Theproposed method can be used to predict the atomic vacancyconcentration distribution and void formation in metals oralloys that are subject to current loading.

IntroductionElectro-migration is the phenomenon that is caused by

atomic migration under the influence of electric current, and isusually also associated with thermal, and stress effects [1]. Itis widely accepted that high current density is the main causethat increases the probability of the collision betweenelectrons and diffusing atoms that result in void and hillockformation at the cathode and the anode respectively. In recentyears, high density packaging (HDP) in electronicsmanufacturing has been increasingly adopted to meet theneeds of miniaturization in electronic products. As a result,current density has increased in conductors with decreasingcross-section area. Moreover, in order to achieve highperformance, current passing through individual solder jointsis approaching 0.4A [2]. Therefore, Electromigration (EM)has become an important issue in the reliability of theelectronic devices.

In a conductor where EM causes void and hillockformation, resistance and joule heating increases. Eventuallythis will cause an open circuit. For the manufacturingindustry, it is important that failures caused by EM can bepredicted. In the 1960s, Black [3-5] proposed a lifetime modelto describe relationship between the mean time to failure andthe current density (Eqn. 1). In this model, E; is the activationenergy, k is the Boltzman constant, T is the temperature, and Ais a material property and structure dependent parameter.

A (·Ea.)·MTF=--;;exp -J':' ia

A EMTF = -nexp (--E.)

J kT

(1)

(2)

This issue was discussed by several authors [7, 8]. Thevalue ofn has been found in the range 1 to 3 [6,9, 10, 11]. Atpresent, there is not a single theoretically reliable value, andsome of experiments that have been used to obtain the valueof n have neglected the thermal effects [12]. In fact, the truemechanism of EM is hard to understand because it happens atthe atomic scale, and it is difficult to separate EM fromthermal migration (TM), stress migration (SM) and otherstructure related effects. Since EM involves many physicalprocesses, describing the EM mathematically and fullyunderstanding its complicated mechanism has become one ofthe hot topics for researchers and the electronics industry.

In recent years, a large amount of work has been carriedout on EM in order to better understand the EM phenomenonand to develop better models that can accurately predict thereliability of electronic components that are exposed to therisk of failures caused by EM [13]. As an important tool,computer modelling methods have become widely used in thisarea [14]. In 1988, A.P. Schwarzenberger et al. [15] used aone-dimensional model to simulate the stress evolution forvalidating their experimental results. Chang et al. [16]simulated the change of the voltage distribution in the solderjoint to predict the site of void formation. Xia and et al. [17]analyzed the motion and evolution of voids due to strain anddiffusion effect using the finite element method (FEM).

In 1986, Shatzkes and Lloyd [18] proposed a onedimensional diffusion-convection EM model and since thencomputer modelling has gradually become more focused onEM under the influence of multiple physical processes. Lee etal [19], modelled both the heat flow and the current density ina solder bump to explore the relation between the combinedeffects of these two factors on the atomic flux. Hieu and Salm[20] analysed the current crowding effect in a solder joint andpredicted the failure location and the lifetime of the solderjoint. Singh et al. [21] simulated the nucleation and initialgrowth of voids in a realistic interconnect test structure. Moreambitious simulation that takes into account all major factorsthat affect EM has also been carried out. For example, Liu etcused the commercial FEM software package ANSYS and aseparate diffusion equation solver [22] to couple electric,thermal, and stress effects. Similarly, Cacho and hiscolleagues used Comsol in their work to solve coupled EMequations [23].

In this paper, a new multi-physics computer modellingmethod is described. This method is based on a tightlycoupled solution of the EM governing equation and it offers a

2011 International Conference on Electronic Packaging Technology and High Density Packaging978-1-4577-1769-7/11/$26.00 ©2011 Crown 448

complete solution from vacancy diffusion to void formationand evolution.

Modelling MethodologyAs mentioned above, EM is affected by many factors that

are difficult to decouple. In most computer simulationresearch work that has been done to date not all of thephysical processes have been taken into account which makesit difficult to compare the results with experimental results.More recently, however, some researchers have carried outEM simulations that include a number of physical processesthat are known to contribute to EM [22, 23]. In this work, aclosely coupled multi-physics modelling method has beenproposed. It can be used predict atomic concentration andvoid formation in metals where EM is affected by electrical,thermal, stress, and geometry factors.

The method is outlined in the flow chart in Fig. 1. Itdescribes the way EM is modelled seamlessly, from electriccurrent prediction to void formation. The model has beenimplemented using the commercial multiphysics softwarepackage PHYSICA [24], which is capable of solving fluidflow, heat transfer, electric field and general diffusionequations simultaneously.

- -D [(VC _IZ-IC1:p

s j ) _.!L- C ,VT+ If] C ,vcr] (4)- :!.-' :!.-' k T e k T'1;!; k T 2;

where

(5)

and.D; is the diffusivity of vacancies, Z" is the effective charge,e is the elementary charge, k is the Boltzmann's constant, Tis

the temperature, E is the electrical field, Qx is the heat oftransport, f is the vacancy relaxation factor, fl is the atomicvolume, (J is the hydrostatic stress, Do is the pre-exponentialfactor and Eais the activation energy.

In Eqn. 4, the terms on the right hand side representcontributions from self-diffusion, electric current, temperaturegradient and stress gradient. In EM analysis, vacancyconcentration CF' rather than atomic concentration Ca is used.Because of the vacancy exchange mechanism, at equilibrium,the relation between the vacancy diffusivity and atomicdiffusivity is given by:

(6)

Where, Da is the diffusivity of atoms. From Eqn. 3, Eqn. 4can also be written as the following diffusion equation:

where

. = Iz>:1 p6!D~: (~) _ Q>: ~~: ('17~) + tnD~: ('170")U k T ut- T':' k T

(7)

(8)

where CFis the atomic concentration, 'F is the total vacancyflux, r is the source/sink term. The vacancy flux vector isshown in Eqn. 4.

Fig. 1: Flow Chart of the EM model

Governing EquationsBased on Fick's laws of diffusion, the governing diffusion-

convection equation for vacancy evolution can be written as:

acp + V·J, = rat ]; (3)

with, u being the drift velocity.

Results and DiscussionsTo test the model, a one dimensional analysis was used.

The computational domain has a length of 100 urn and it wasdivided into 50 elements with constant cross-section area. Anelectric potential difference was applied at the two ends of thedomain generating a uniform electric filed and a constantcurrent. The EM drift velocity is 0.214 m/s if the only EMeffect is the electric current and the parameters in Table I areused. In order to test the effect of temperature gradient, alinear temperature profile was imposed. Similarly, a linearhydrostatic stress profile was created to test the stress gradienteffect. The values of the temperature and stress gradients are+0.107 CO/m and +0.107 MPa/m. Fig. 2 shows the vacancyconcentration at t=800 s for simulations with electric effectonly, electric and thermal effects, and all three effectsrespectively. The results show that the temperature and stressgradient have both had an impact on diffusion.

2011 International Conference on Electronic Packaging Technology and High Density Packaging 449

Ea o.o-v [27]

z* -5.0 [28]

P 1.69xl0-6Ocm [26]

J 2MA/cm2 -L IOO11m -T 573K -

Fig. 2: The vacancy concentration at t=800 s forsimulations with electric effect only, electric and thermal

effects, and all three effects.

Moreover, we have also solved another ID EM problemthat has an analytic solution derived by R.L. de Orio and hiscolleagues [14]. The governing equation is Eqn. 3 without thethermal and stress terms. The boundary conditions are shownin Eqn. 9 which states that perfect blocking boundaryconditions are imposed at both ends of the ID domain. Thesolution is expressed as Eqn. 10 [25].

/v(O,t) = t.:-L,t) = 0 (9)

where L is the length of the line.

c~:Cx.,iJ A "\'-00 _ [ D~: a x-c- = 0 - Ln=l An exp -Bn 2 t + --)

~:o L 2 L(10)

Fig. 3: Vacancy distribution at (a) 100 s, and (b) 800 s.

where

Iz:< lepjLa=---

kT(11)

and

Ao = cr( . exp (a ~) ,1-e.xp -aJ L

(12)

An =16ITa

2[1-(-1) n eX"P(~)) . ( x) 2nIT ( x)

[., 4 ., .,-., [Sln nn- + - COS nn- ] (13)

a..:. + n..:. IT":' J":' L a L

and Fig. 4: The vacancy concentration along the X-axis at

different times

Table I: Parameters used in the calculations

Fig. 3 shows the simulation results using the parametersgiven by Table I and a=3.4. In Fig. 4, simulation results atdifferent time are consistent with the results of analyticalsolution.

At present, there is no general analytical solution forEqn. 3 with all the terms included. In order to validate themodel, work is under way to compare the modelling resultswith experimental results. Fig. 5 shows the test specimengeometry. SnAg solder is plated onto a glass substrate. Thethickness of the solder layer is 0.55 urn, The main objective ofthe experiment is to study the mechanisms of EM at thenanometer scale and to compare with simulation results. Fig. 6shows some early test results. The voids and hillocks areclearly seen in the picture.

(14)

[26]

ReferenceValueParameter

DvO


(a) (b)

damascene Cu/Si02 structures. However, for the two E;values of 0.9 and 0.7 eV, the drift velocities are very different(Table II and Fig. 7). Similarly, drift velocity is also sensitiveto changes in temperature (Fig. 8). Our future works includesan attempt to use modelling and experimental results to obtainthe E; value for the material that is used in our experiments.

Table II: The drift velocity and the activation energy atT=310 K.

Fig. 5: The geometry of the experiment

Fig. 6: From top to bottom: hillock at the anode, area inthe middle of strip, and the void at cathode.

Material Properties and ExperimentsThe activation E; has a very strong influence on the EM

process. Liniger et al [27] measured the E; values for anumber of materials using experiments and Black's model.The value of E; for Cu was found to be 0.9±0.1 eV, which isthought to be consistent with Park and Vook's value of 0.79eV [29] and the 0.7-0.9 eV obtained by Hu et al [30] for dual

E (eV) 0.9 0.7a

V (m/s) 2.84 E-15 5.07E-12

vtjtm/hour) 1.0224E-5 1.8252E-2v(l urn/time) 11.2 years 2.28 days

(54.8 hours)

Fig. 7: The drift velocity as a function of Ea. T=310K

Fig. 8: The drift velocity as a function of temperature. Ea=0.9

eVe

ConclusionsA multi-physics modelling framework has been proposed

and implemented using the software package PHYSICA. Theprocesses of diffusion, electric current, temperature gradientand stress gradient in electromigration can be solvedsimultaneously. Early results have shown good agreement


with an analytical solution. Further work will be carried out inorder to further validate the methodology using experimentalresults.

AcknowledgementXiaoxin Zhu would like to thank the University of

Greenwich and Sha Xu would like to thank HK RGC GRF(Project No.9041486), CityU Research Activity Funds forfinancing this research work.

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