Electromigration and mechanical stress

ELSEVIER

Microelectronic Engineering 49 (1999) 51-64

www.elsevier.nl/ locate/mee Electromigration and M e c h a n i c a l Stress J.R. Lloyd Lloyd Technology Associates, Inc. PO Box 194, Stow MA 01775-0194, USA The state of the art in the understanding of electromigration failure and mechanical stress is reviewed. First, the important concepts are introduced the effects of thermal stress and the presence of a refractory underlay is shown. Following a simplified way of qualitatively assessing the effect of various geometries on electromigration is proposed.

1.0 Introduction Electromigration is the major wear-out failure mechanism in thin film conductors used in microelectronics. As such, it has been the subject of intense study in industrial laboratories for more than three decades. (1) Our research into electromigration induced failure has led us to a reasonably good level of understanding by this time. This has brought us to picture electromigration as the consequence of mechanical stress gradients generated to oppose the electromigration driving force.

2.0 The electromigration driving force Electromigration is best characterized by the Einstein equation for diffusion in the presence of a driving force, F,

where e is the electronic charge, p is the electrical resistivity, j is the current density and z* is known as the effective valence or the effective charge, although it is neither a charge nor a valence, z* is an interesting quantity that reflects the direction and the magnitude of the momentum exchange between diffusing metal atoms and conduction electrons that is responsible for electromigration. The solid state physics explanation for the driving force is beyond the scope of this paper, but is readily available in the literature. (2-13) Where electromigration becomes interesting is that the driving force in eqn. (2) is not the only one that is operating on the metal system while carrying current. The real driving force for diffusion of any material is the gradient in the chemical potential.

DCF

J - - -

kT

-

DC

V/z

(3)

kT

--9

--' D F C J = - kT

(1)

where J is the mass transport D is the diffusion coefficient, C is the concentration of diffusing atoms, and kT is the average thermal energy. For electromigration; F:Z

*epj

(2)

The chemical potential gradient has many components, electromigration is but one of them. Early in the study of electromigration, investigators were puzzled by the failure kinetics. Eqn. (2) would suggest that failure should be dependent on the applied current density, j, and that lifetime should be directly dependent on 1/j. However, more often than not, the failure kinetics obeyed a different relation that has become known as "Black's Law". (14)

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J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64

Black's law was the major results of the early work of J.R. Black of Motorola who performed the first comprehensive studies of electromigration failure. It relates the median time to failure of an ensemble of samples as a function of current density and temperature;

t o:A exp( ) 2

AH

(4)

where AH is the activation energy for failure and A is treated as a geometry dependent constant. The discrepancy between the observed behavior described by eqn. (4) and the theoretical understanding described in eqn. (2) was resolved initially by considering that the chemical potential contained a concentration gradient term;

J

=

D( z*epj C - OC [ KT dx

(5)

where C is the concentration of diffusion atoms, or vacancies. (15) In the presence of a blocking boundary condition, J(0,t) =0, the continuity equation;

bJ

dC _ D( bzc

Tx-Z

lax

z* ep j bC ]

-ff

)

(6)

produces a solution where the time to achieve a specific concentration is proportional to the inverse square of the current density.

tso = A'

exp

reasonable values for parameters were inserted into the relation for the pre-exponential constant, the lifetimes appeared too short. Additionally, there seemed to be no way to incorporate the effect of the stress gradient on electromigration. Attempts were made to include stress by using the equilibrium between stress and vacancy concentration, but the results were not very satisfying.

(7)

This solution has the same current dependence as Black's Law, eqn.(4), but with a slightly different temperature dependence. Activation energies calculated with eqn. ( 6 ) are slightly higher that those calculated with eqn. (4), but are within experimental error. Although this treatment resolved the issue of the inverse square dependence of the electromigration lifetime, it was not a complete description. When

The problem was resolved in part a few years later with the work of Korhonen et. al. at Cornell. (16) Here it was realized that excess vacancies could not be supported in a thin film and their eventual annihilation would produce a tensile stress. The resulting stress gradient would then act as the driving force opposing electromigration and not the entropy gradient treated earlier. After simplification of the picture to allow an analytical approach, a reasonable solution to the diffusion equation was obtained that confirmed the 1/j 2 dependence for nucleation dependent lifetimes. In addition, a more correct depiction of the process of vacancy production and annihilation resulted in values for the pre-exponential that better agreed with experimental results. Later efforts by Clement (17) have further improved the theoretical picture of electromigration stress generation and subsequent failure. In all cases, the analytical expressions for deceptively simple looking equations and boundary conditions have proved to be unexpectedly difficult. On the other hand, the steady state solutions, in most cases, are nearly trivial. After introducing some important concepts, we will use this fact for a simpler approach to qualitatively predicting the electromigration behavior of various configurations of conductor geometry that can be of use to designers and reliability engineers.

3.0 Blech Effects There are three important concepts that need to be understood that will be presented collectively under the heading of "Blech Effects". These are the "Blech Product", the "Blech Length" and the "Blech Condition". They are all related to the concept of stress gradients opposing the electromigration mass flow, occasionally referred to as the "back stress". These were discovered more than 20 years ago by I.A. Blech and reported in a series of short papers that should be required reading. (18-20)

53

J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64 3.1 The Blech Product In an effort to measure the electromigration induced drift velocity in metallic thin films, Blech conceived of a clever experiment. (18-20) He began by depositing islands of conductor (Au and AI ) onto a resistive refractory (TIN) underlay. Since the resistivity of the metal was substantially less than the refractory, most of the current would be contained in the metal. The original idea was that since the ends of the conductor were free to move, the drift velocity, VD. could be measured as a function of current and temperature according to

Dz * ep j vD -

kT

(8)

The results, however, were unexpected. For long lines, the upstream (with respect to electron flow) appeared to follow eqn. (5) for a while, but the downstream side of the island didn't move. Instead, extrusions were generated on the downstream end. Since only the upstream side of the island moved, the island become smaller with time. When the island shrunk to a particular size, the upstream end stopped moving and electromigration essentially came to a halt. This led to the relationship

ABt~c~ = j x l 8

(9)

where ABlech is called the Blech Product and 1B is the length of the island. The value for ABlech was found to be a function of temperature and to be different for conductors covered by a glass passivation as compared to uncovered films. Aalec h was found to be on the order of 2000 Amp/cm. la has become known as the Blech length.

3.2 The Blech Length

should be substantially longer. If the test structure is shorter than the Blech length, the part will never fail. This becomes particularly important when studying failure in vias or contacts. Unfortunately, it had become common practice in many laboratories to design test structures with many vias in series, connected by short links. The links were as short as possible to get the maximum number of vias for a given area of test structure. The Blech length effect, however, does not permit this. If the links are less than la, there can be no failure in any vias connected by the short links. A false sense of security will result from the use of such a design. On another note, eqn.(6) suggests that at the lower current densities used in normal operation, the Blech length is substantially longer. At the traditional military specification limit of lxl05 Amp/cm 2, IB > 100 ktm. Therefore, any conductor shorter than this will not fail. It also suggests that for short lines, higher current densities can be permitted without concern for electromigration failure. As it turns out, this optimism is only partially realized, as we will see in our discussion of thermal stresses.

3.3 The Blech Condition The Blech effect was understood by realizing that a component of the chemical potential gradient in eqn. (3) is a gradient in the mechanical stress. (21) This has been known for many years as being responsible for a phenomenon known as diffusive creep. (22) When the stress gradient and electromigration are both considered, eqn. (3) becomes

For Al31ech on the order of 2000, at a typical stress current densities of 2x106 Amp/cm 2 la is 10~tm. Therefore, any line less than 10~tm long will not experience electromigration. Such lines are immortal with respect to electromigration failure. This interesting observation has at least two very important practical consequences.

where ~ ts a quantity known as the activation volume and (~ is the hydrostatic component of the stress.

For test structure design, the length of the conductor must be at least as long as IB, and in fact

If the quantity in the parentheses in eqn.(10) vanishes, electromigration also vanishes. This will

=

z*epj kT

(lO)

~x

J.R. Lloyd I Microelectronic Engineering 49 (1999) 51-64

54

occur when the driving force from the sstress gradient in the second term exactly equals the electromigration driving force. The stress gradient at this steady state condition becomes;

~cr

z * ep j

bx

f2

(10a)

which is known as the Blech Condition. Integrating over the length of the conductor, we obtain the following relation for the stress as a function of x, o(x);

O'(X) = O"o +

z* e p j x

(lOb)

where G0 is the stress at x=0. The stress is seen to be linear in x. The value for x that corresponds to the maximum stress that the conductor can withstand, ~m, is the Blech length, IB

IB =

(10c)

z*epj The extension to the Blech Product is obvious. The preceding arguments are the basis for the presently accepted view of electromigration failure. In the presence of a flux divergence, a stress gradient is induced that opposes the electromigration force. If the stress is allowed to rise to a critical point, the conductor stripe will form a void to relieve the stress. Strictly the arguments above do not necessarily describe void formation. There is no argument as to the sign of the stress that is generated. There is the tacit assumption in the discussion that we were talking about tensile stresses and void formation, but there is nothing in the mathematics that will preclude the formation of compressive stresses and extrusions. In fact, both types of failure occur and have been observed in the same sample, often coexisting in the same conductor line. (23) For an extrusion to form, the failure criterion will not be the tensile stress required to form a void. In uncovered conductors, the compressive stress required to form an extrusion will be the criterion.

This stress is rather low, especially at elevated testing temperatures. In conductors covered by passivation layer, such as an interlevel dielectric, the failure criterion will be the stress required to cause a failure in the passivation layer. Compressive stresses in the metal will produce tensile stresses in the passivation in a similar fashion to that in a thick walled tube. These hoop stresses are always tensile and always greater than the compressive stress in the enclosed metal. The thicker the passivation layer, the lower the induced tensile stresses for a given compressive stress, so that thick passivation layers produce conductors more resistant to extrusions than those under thin passivation layers. (24) In Blech's early experiments, failure was really due to achieving the threshold for extrusion and not that for void formation. The upstream edge of the conductor acted like a pre-existing void. Therefore, none had to be nucleated since there was a free surface present at all times at near ~ = 0. When he performed island experiments on passivated conductors, the observed Blech Product increased accordingly. Not only that, but when the experiment was completed and the metal conductor was allowed to relax without current, the edge was seen to migrate back in the direction of its original position. This was a clear demonstration of the effect of the stress gradient acting as a driving force for mass transport.

4.0 Origin of stress 4.1 Korhonen Approximation The origin of the mechanical stress was first suggested by Blech in an early paper, but later elaborated in a more formal way by Korhonen et. al. In this paper, it was realized that the vacancy diffusion equation of Shatzkes and Lloyd and subsequent papers, (15, 25, 26) was flawed in the sense that a supersaturation of vacancies cannot be supported in a thin film as described. Korhonen (16) suggested that vacancies would annihilate at convenient sinks and in doing so would produce a stress. A simple expression of Hooke's Law

/3

(11)

J.R. Lloyd I Microelectronic Engineering 49 (1999) 51-64 where Cv is the vacancy concentration and 13 is the bulk elastic modulus was invoked that led, after simplifying assumptions, to the expression;

Oty

O ( Oty

z * ep j

(12)

~t :D'~x~x-x + k-T )

2

~Cr=afOCrOt ~Ox +bOerox+COx2b2cr

(13)

for which an analytical solution is not available. Although the problem has these somewhat unsatisfying shortcomings, the general principles involved in calculating the stress are correct and the solution for the failure time

tso = A ' 4 e x p ( AH I j ~kT)

4.2 Clement Formulation

In this treatment, (17) Clement considered vacancy diffusion, but added a sink/source term, 7.

OJ_ dC _ D(02C z*epj OC) ax

where D~ is a reduced diffusion coefficient. The problem with eqn,(9) is that it expressly states that for a change in the stress to occur, there must be a divergence in the current density. This is clearly in error. This problem arises from the simplifying assumptions that were made in order to obtain an analytical solution. One of them that the vacancy concentration is constant. However, since it is the annihilation of vacancies that is responsible for the stress in the first place, this assumption is selfinconsistent with the model being proposed. (27) The correct expression is the non-linear partial differential equation

(14)

has the required current density dependence to agree with experiment. Calculations of the activation energy using this model lie between those using Black's equation, eqn (4), and the model of Shatzkes and Lloyd, eqn. ( 6 ) . The problems with this model should not be detract from the major contributions in understanding of the source of the stress that the introduction of this model has made. However, it is useful to pursue more exact solutions as in the following.

55

l-a-

kT

ax +r (15)

The sink term, 7, is usually expressed as

]/-

C - Ce - -

(16)

"t" where Ce is the concentration of vacancies in thermal equilibrium and "c is the "lifetime" of a vacancy in the presence of the sink. The strain was calculated as in the Korhonen model. Interestingly, this solution arrives at yet another, slightly different, equation for electromigration lifetime. ,.r

3

,50 = ,,i --~-exp[--~- j

(17)

Experimentally, it would be exceedingly difficult if not impossible to differentiate among the four expressions for lifetime, eqns, (4, 6 , 9 , 10b). Of course, the calculated activation energy will vary somewhat due to the different temperature dependencies of the pre-exponential expressions. However, lifetime extrapolations differ by less than a factor of two from most stress to use conditions. For this reason, the traditional use of the Black Equation is not harmful, especially in view of typical lot to lot variations. 5.0 T h e r m a l S t r e s s

The thermal coefficient of expansion (TCE) for metal conductors is much greater than it is for Si or any of the materials used for interlevel dielectric or passivation. As an example, AI has a TCE of -23ppm/C, Cu is ~17ppm/C whereas SiO2 is ~3ppm/C. This becomes important when one considers the processing history of a typical IC, where the passivation layer may be applied at an

J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64

56

elevated temperature (-400C). If there is good adhesion between the metal and the passivation, a thermal strain will be created in the metal upon cooling. This "thermal strain", eth, is a function of the difference in the TCE of the dielectric and the metal, Aa, and the temperature difference between the application of the dielectric and the operating temperature, AT.

eth = A a A T

(18)

For a typical values of Aa (-20 ppm/C ) and AT (-200C) we can expect thermal strains approaching 0.4%. This strain is unusual, and sinister, in that it is primarily a hydrostatic tension. A hydrostatic stress cannot be relieved by slip or yielding that is not accompanied by a change in volume. Therefore, the only way this stress can be relieved is by void formation and growth. Of interest to electromigration failure is that since void formation depends on reaching a critical stress for the void to nucleate, the presence of a high tensile stress will reduce this threshold, making it easier to form a void. This will also shorten the "Blech Length" for void formation. Substituting the thermal stress in eqn. (7c) we obtain for 1B

/

IB = Crm

EAaAT 1 - 19

) ° z

ep j

*

(19)

where E is the elastic modulus and v is Poisson's ratio. As the thermal stress is increased, IB becomes shorter and shorter. This implies that as the thermal stress increases, electromigration lifetime, if it is determined by void nucleation, will be correspondingly reduced. (28) When the thermal stress reaches the maximum stress, 1~ vanishes, we have no Blech length and void formation even without current. This condition is familiar under the title "stress voiding". Therefore, the possibility of using the Blech Length as a means to achieve increased reliability is tempered. The effective Blech Length in the presence of the inevitable tensile thermal stresses is much reduced as compared to that calculated from some experiments. We can no longer attempt to design circuits where all conductors are less than IB

without considering the effects of the thermal stress that can reduce la to zero. In modern IC technology, we have a way out of this dilemma, but have replaced it with another. In most contemporary IC processes, a barrier layer is placed under the metal conductor for various reasons. One of the beneficial effects of this boundary layer is that if a small void were to form, the barrier material, usually a refractory metal or compound, will act as a redundant conductor. If a small void were to form, the barrier metal can conduct electricity. A common barrier material is TiN, which is immune to void formation for extremely long times at high current density, effectively eliminating open circuits as a failure mode. Although voids will not cause catastrophic failure in the form of opens, the resistance will naturally be increased as the void forms and grows. The time to failure now is not the time required to nucleate a void, but, the time for the void to grow to a size where the associated increase in resistance will cause circuit failure must be included. The thermal stress becomes of critical importance in the reliability of a metal system incorporating a redundant barrier layer. For one, the failure kinetics will change radically. If the thermal stress is sufficiently high that the Blech Length is very small or vanishes, void nucleation will be rapid. Void growth does not depend on the square of the current density, but will depend roughly on the current density to the first power. Therefore, for failure that depends on void growth rather than void nucleation, failure kinetics will change from n=2 to n=l. Experimental failure times will have to be significantly longer for any given accelerated current density to insure reliability at use conditions. Naturally with this bad news, Mother Nature provides some good news.., maybe. In a metallization with a barrier layer and thermal stress, there will be a maximum void size that can be attained. (29) If the resistance increase that is associated with this maximum void size is not "fatal" to the circuit, we will not have a failure. Unfortunately many of the more heavily unidirectionally stressed lines are long, and longer lines will suffer larger voids.

J.R. Lloyd I Microelectronic Engineering 49 (1999) 51-64 The size of the maxim void size and, therefore the maximum resistance increase depends on the sum of the thermal and the electromigration induced strains. Initially a stress is built up to the point where the void nucleates. Thermal stress makes this easier and perhaps a stress void may be present even before any current is passed. Following void nucleation, the stress at the void site vanishes (there cannot be stress inside a void or on a void surface unless it is a gas bubble). This produces a steep stress gradient that will promote rapid diffusion away from the void. The void grows rapidly until this stress gradient is reduced. Following this "inflationary growth period", which accommodates much of the thermal strain, the stress gradient is re-established. The strain associated with this stress gradient contributes to the void size. The ultimate thermal void size will be simply

Alth = 31ActAT

(20)

where AI is the change in the length, 1, of a conductor of arbitrary cross section assuming the void consumes the entire line. The factor of 3 in eqn.(13) is due to the fact that the length change is linear and the thermal strain is three dimensional. Eqn. (13) suggests some remarkably large voids can form due to the thermal strain. Given typical values for IC conductor lines (Act = 20 ppm/C, AT = 200C, 1 = 100 ~tm), thermal voids longer than lktm can be expected. In a submicron line, this can produce significant resistance increases. When the electromigration stress gradient is reestablished there will be additional void growth due to the displacement from the electromigrationinduced strain. This can be calculated by integrating eqn.(7 ) over the length of the conductor and substituting the strain for the stress.

Alem

~ Z * ep jx dx = z * ep jl 2 =

~E

2~E

(2I)

l

The calculated electromigration displacement in a 100 I.tm long line is comparable to the thermal displacement. For longer lines the displacement is dominated by electromigration effects and for shorter lines dominated by thermal effects. Keep in

57

mind, however, that electromigration is still responsible for the thermal displacement in that it forces the thermal strain to one end and, furthermore, is primarily responsible for the increase in stress that led to void nucleation.

Figure 1 Large void under a W plug following an electromigration test. The void is -4 lam long. The conductor line containing the void was -200 ~m long. Thermal strain is responsible for at least half the void volume. There were no extrusions visible.

6.0 Steady State Stress Diagrams The solutions to the boundary value problems describing electromigration are deceptively difficult. Although they look rather straightforward, the solution is often complex, if available at all. The steady state solutions, however are relatively trivial. Although the steady state is never reached, much can be learned about the potential performance of a configuration and much can be explained, even semi-quantitatively. The reason for this is that the chemical potential of vacancies, ktt~, can be expressed as;

l.t~ = k T l n ( C~ l

tc<.)

(22)

where Cv is the vacancy concentration and Ceq is the concentration in thermal equilibrium with the steady state stress, given by;

C v = Cvo e x p

(23)

J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64

58

where C~0 is the vacancy concentration in a stress free metal. Substituting eqn(23) into eqn (22) produces the result that the chemical potential is represented by the difference between the stress at time, t and the steady state value, ~ (24)

/,t(t) : f 2 [ O ' ( t ) - O'= ]

Therefore, by studying the steady state stress profile, much can be learned about the performance of a particular configuration. In addition, an estimate of the time dependent behavior can be obtained by realizing that the driving force for diffusion is the gradient in the chemical potential as seen in eqn. (3). The failure times will be thermally activated according to the diffusion mechanism and linear in the difference between the initial and the steady state value of the stress. This is a very simple way to understand electromigration failure.

6.1 J--O Boundary Conditions 6.1.1 Constant current density with no initial thermal stress First let us examine the case of blocking boundaries on each side of a metal stripe. This is can be considered one of the two "normal" configurations of a conductor in an IC. The first configuration will be that when the conductor is bounded on each end by a contact to Si or to a W via. (J(0)=J(1)=0) In the second, the conductor will be bounded on one side by the contact or via and on the other by a bond pad. For the perfectly blocking boundary conditions before void formation, we can see in Figure 2, a symmetrical stress profile with the maximum tensile stress numerically equal to the maximum compressive stress. . i - o~ ,~o-f°

Failure Stress (Extrusion)

.~-'° /., I° .° I

°

~°

.-

,~°°

/"

Stress ~,°~ °,~° °I""

,t. ~"

Failure Stress (Void)

Distance from Divergence

Steady State Stress Before Void Formation

.....................

Steady State Stress After Void Formation

Figure 2 Steady states stress profile for the case of the J (0) = J(l) = 0 boundary condition (Case I) before void formation and after the void has formed. Following void formation the boundary condition changes to the ~(0) = J(1) = 0 boundary condition, (Case II) the same as for a bond pad at x=0. Note that the maximum compressive stress doubles after the void is formed. Thermal stresses are ignored here. The tick mark on the x-axis represents the length of the conductor between contacts or vias. Note that the compressive failure stress is assumed greater than the tensile failure stress.

J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64 In Figure 2 we have assumed that the compressive stress required to form an extrusion is numerically greater than the stress required to form a void. It should be mentioned that this is an assumption and open to argument, and would certainly be a function of the choice of passivation or interlevel dielectric material. In this case, we have allowed the maximum tensile stress to just equal the threshold for void formation. The slope of the stress line is directly proportional to the current density, so that if we wanted to avoid damage we would need to reduce the current density. The Blech condition could be realized at a lower stress, electromigration would cease and the conductor would be immortal. At higher current density, the maximum steady state stress would exceed the threshold and we would eventually form a void. Since the driving force for diffusion is given by eqn. (24), as we increase the current density the time to reach the threshold stress is reduced accordingly. In fact, it has been shown that the time to reach any arbitrary a < ¢~max is inversely proportional the square of the current density. (15-17) Note also that, since we have assumed that the threshold compressive stress is greater than the threshold tensile stress, extrusions would never form in a sample of this length, denoted by the tick mark on the x-axis. Let us now assume that a void did form. (Of course, it could be argued that for the configuration shown above a void would never form if (~max were exactly the threshold stress, since the driving force vanishes as O'max is approached. However, in this discussion, we will allow Omax to just exceed the threshold.) Once the void forms, the stress at x=0 must vanish. The void surface cannot support a normal stress, so there can be no hydrostatic component of the stress. The sequence of events shown in Figure 3 is key to what happens. In Fig. 3b, just as the void forms, the stress assumes a profile near the void where the driving force for mass flow is suddenly in the direction of electromigration instead of opposing it as it did previously. This results in a rapid increase in void volume initially that slows as the stress gradient is reduced, eventually stopping when the new steady state in Fig. 3c is approached.

59

Most interesting is the observation that the maximum compressive stress at the opposite side of the conductor increases significantly when the void forms. In fact it exactly doubles. In the case shown in Fig. 2 the maximum stress was below the extrusion threshold before void formation, but exceeded it after the void formed. This produces a seemingly paradoxical situation where the extrusion cannot form until there is a void. This suggests that extrusions will follow, not precede voids as is usually assumed.

J Fig. 3a Steady state stress before void formation

J Fig.3b

Instantaneous stress following void formation

Fig. 3c [ Steady states stress after void formation

Fig.3 Sequence of events during void formation.

I

J.R. Lloyd I Microelectronic Engineering 49 (1999) 51-64

6O 6.1.2

Thermal Stress

The effect of a thermal stress is to shift the axis of the steady state stress profile by the amount of the thermal stress. The stress dipole will now be centered around O'th rather than around zero. The first effect of this shift is to make it easier to form a void. Electromigration needs to provide correspondingly less additional stress to form a void. This also will have the effect of reducing the Blech Product or shortening the Blech Length.

lB =

( o r - Crib ) ~

Fig. 4a Steady state stress with thermal stress denoted by the dashed line

(25)

z*epj One of the effects of the thermal stress can be seen in Fig. 4b. Following void nucleation, the driving force for void growth is greater than in the absence of thermal stress. In addition, since the steady state stress after void formation is the same whether there was thermal stress or not, the eventual void is much larger. The void size can be visualized by the difference in the area under the curve between Fig. 4c and Fig. 4a. (Or Fig. 3c - Fig. 3a for the stress free case). We also see how the presence of tensile thermal stresses will have little effect on the eventual development of an extrusion following the formation of a void. 6.2 a = 0 Boundary

J

Fig. 4b Instantaneous stress followingvoid formation

Conditions

The ~ = 0 boundary condition corresponds to a bond pad or an open surface like a void. One ended er= 0 boundary conditions are common in IC designs, such as a conductor connecting a bond pad to a via or contact. This is covered in the previous section. The case where both sides of the conductor have the ~ = 0 boundary, however, does not occur in "real life" but is common in test structures. The NIST standard and many other test structures possess this. Therefore it is important to understand failure in this configuration, since so much data has been generated under this condition.

Fig. 4c I Steady states stress after void formation

Figure 4 Effect of thermal stress

]

J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64

For a continuous film with no variations in the rate of mass transport, the steady state stress state is ~=0everywhere. If there is an initial thermal stress, the stress will decay until the yield stress is reached. This is a decidedly uninteresting state. It is completely unlike anything that happens in real life. If this were truly the case, there would be no failures, under any conditions. The famous experiments of d'Heurle where pure AI single crystals were stressed for ~ I 0 years under conditions where polycrystalline A1 survived for only a day without failure. (30)

61

Fig. 5a Schematic of grain "cluster" producing a divergence in mass flow

However, we know that failure is possible in NIST and similar structures. We also know that failure in these structures is highly dependent on the microstructure, which can be understood with the help of the steady state stress profiles in Fig. 5. Here we will assume that we have a conductor possessing a "near bamboo" structure, with no continuous grain boundary diffusion pathway in the direction of electron flow. Although a continuous pathway along the entire length of the conductor line does not exist, we will assume that there small grained regions, called "grain clusters", where a network of grain boundary diffusion pathways exist over a limited distance. Keep in mind that what we will be saying here will pertain to any region where is a change in the atomic (or vacancy) mobility. It can be for any reason, not just the cluster example we have chosen here. The clusters are, however, a common example. With the ~=0 boundary condition imposed, an interesting thing develops. In Fig.5 we can see the effects of the location of the flux divergence. In the cluster region, the diffusion is much faster than in the adjacent bamboo regions. Therefore, a steady state profile in the cluster region will be established. Since the boundary of the cluster is not a perfect block (J~0), there will be flux through the boundary. In order for the steady state to be realized over the whole length of the conductor, a stress profile will be established outside the cluster. If the mobility on each side of the cluster is identical, for instance from interfacial or lattice diffusion, the slopes of the stress gradients must be equal. If they were not, the steady state would not be established since there would be a net gain or loss of material with time in the cluster

Fig, 5b Steady state stress with cluster in the middle of the conductor

I

Fig 5c Steady state stress with cluster on the left

J

Steady state stressFig. with5dcluster on the right

J

Fig.5 Effect of flux divergence on steady state stress distribution in a conductor line with a=0 boundary conditions. A grain cluster in an otherwise bamboo line was chosen as an example, but any flux divergence will have the same effect

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J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64

region. (31) In order for this to occur, the absolute values of the stress at each end of the cluster would depend strongly on the position of the cluster with respect to the bond pads at ~=0. If the cluster were in the middle of the line, the symmetric stress dipole would often not present a reliability problem. If however, if it were to be located at either end, in order for the steady state to be achieved the maximum stress would be higher. The type of stress would depend on the current flow. If the cluster were upstream, a high tensile stress would be developed, whereas if it were located on the downstream side a high compressive stress would be developed. We could therefore expect to see more voids near the cathode and more extrusions near the anode in a typical NIST type structure.

I

Fig. 6a Schematic of narrow line portion or incomplete step coverage

,-"

I''" Note even for J=0 boundary conditions, something similar would occur. If there were a cluster, the steady state profile would be rapidly established. This would promote the formation of a stress gradient that would work in the direction of the electromigration flow and would perhaps increase the flux from the J=0 boundary. Thus, a cluster of any size could hasten failure at a contact some distance away. (32) 6.2 Step Coverage, Constrictions and Wide to Narrow Transitions. 6.2.1 J (0) = J(l) = 0 boundary conditions The effects of step coverage and wide to narrow transitions can be significant, primarily because their effect can be felt even if they do not produce a flux divergence, however, there needs to be a flux divergence present somewhere. Let us assume no flux divergence in either a fine grained stripe with many diffusion paths, a perfect bamboo or a single crystal with a single interfacial or lattice diffusion pathway. If we have t~(0)=~(1)=0 boundary conditions, there would be no failure. The steady state value of the stress is identically zero everywhere. Given a J=0 boundary condition, however, the picture completely changes. If the line has a constriction, or if there is a step coverage problem that raises the current density, the corresponding steady state stress profile will reflect this.

I

Fig. 6b Stead Y state stress corresponding to Fig. 6a

¢~SJS~ lII Fig. 6c Steady state stress with narrow portioned shifted upstream

Fig.6 The effect of a narrow portion or incomplete step coverage on a continuous conductor with J(0)=J(1)=0 boundary conditions. An examination of Fig.6 shows that once again the position of the "defect" important. If the volume of the conductor on either side of the defect is the same, as in Fig. 6b, the area under the curve must sum to zero. With no flux into or out of the conductor, there can be no net change in the amount of material. If the volume is not the same, then the stress will not be symmetrical. In Fig. 6c, we have a situation where, due to location, the upstream side has a smaller volume then the downstream side. This will produce a higher tensile stress in the

J.R. Lloyd / Microelectronic Engineering 49 (1999) 51-64 smaller volume. A considerable amount of material must be removed to bring the stress in the larger volume downstream side up. The larger the relative volume of the downstream side, the higher the stress in the upstream side. In this light the consequences of poor step coverage are clearly seen. Although a stress profile may be easily developed to counter the added driving force in a short length of conductor, the effect due to the long range steady state stress can be considerable.

6.2.2 J(O) =0, a(l) =0 type boundary conditions The configuration near a bond pad into a contact or a via is significantly different from the prior one. (Sec. 6.2.1) The effect of the defect also depends on the location of the defect or transition and additionally on the direction of current flow. The main difference between these configurations lies in the lack of the restriction on displacement. With a bond pad (G = 0) on one side, the constant volume and the area under the curve will not integrate to zero. Therefore, there is a significant difference with respect to current flow. Let us examine the differences of current flowing from a wide to a narrow line and the converse. With current flow out of a contact into a line terminating in a bond pad, we have J(0) =0, ~(1)= 0 boundary condition. In Fig. 7 the steady state stress for current flowing from wide to narrow denoted by the solid line whereas the stress for current flowing from a narrow to a wide line is shown by the dashed line. Due to the ~ = 0 boundary condition, the maximum stress in each case must be the same, however, the material displaced to produce the stress is considerably different. Clearly the narrow to wide case requires more material displacement than the wide to narrow. If we consider transition from lines of equal length to half (or double) the width, the narrow to wide case represents twice the material displacement than the wide to narrow. When a void eventually does form, we have a particularly insidious condition. Since the void cannot support stress, the previous J=0 boundary condition immediately becomes ~=0. Not only will all the strain be "dumped" into the void as we saw earlier, but now there is no opportunity for a back

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stress to develop. The lack of a reservoir of material will enable the void to grow without restriction. Resistance will increase without saturation until almost the entire conductor is emptied into the bond pad. Note that the previous situation is also present where stress voids are initially present in the structure. In a stress voided line there is no opportunity for back flow in lines that have an initial ~--'0 boundary condition. There will be no back stress and n= 1kinetics will prevail. Current direction obviously makes a difference here. If the current is flowing from the bond pad to the contact, voids will not form and a steady state stress will be established opposing electromigration and the failure of metal near the bond pad is highly unlikely. The implication for design rules is obvious.

7.0 Concluding Remark The present appreciation of the role of stress in electromigration failure has enabled us to come to a more complete understanding of the behavior of thin films carrying high current densities. Electromigration can be considered a form of diffusional creep. Although the time dependent solutions to the transport equation are often difficult to obtain, a simple look at the easily obtainable steady state stress distributions can illustrate the essential character of electromigration phenomena.

sssssSSSSpsssS Fig. 7 Steady state stress in a wide to narrow transition with a a =0 boundary condition.

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