Stress evolution and drift kinetics in confined metal lines

Microelectronic Engineering 50 (2000) 341–347 www.elsevier.nl / locate / mee

Stress evolution and drift kinetics in confined metal lines S.A. Chizhik*, A.A. Matvienko, A.A. Sidelnikov Institute of Solid State Chemistry, Kutateladze 18, Novosibirsk, Russia Abstract A model describing stresses and vacancy system evolution and also drift phenomena caused by electromigration (EM) in metal interconnectors has been proposed. It is shown that the drift kinetics has three main stages, i.e., induction period, quasi-stationary and stationary state. Induction period is characterised by an increase in the stress at the ends of a conductor; drift is absent. Drift begins when the yield limit is achieved at the cathode edge of the conductor. Drift rate increases, reaches a maximum and then decreases down to the stationary value. Characteristic times for all the stages are determined. Also, the effect of the length of the conductor on drift kinetics is analysed.  2000 Elsevier Science B.V. All rights reserved. Keywords: Electromigration; Stress evolution; Drift kinetics

1. Introduction A unique situation in the understanding of EM processes has been developed. On the one hand, there are excellent works presenting the models that describe the evolution of stresses during EM (e.g., Ref. [3]). However, these works do not consider the evolution of stress under drift conditions and the influence of stress evolution on drift kinetics. On the other hand, there are many works studying drift (e.g., Ref. [1,2]). In order to analyse stationary drift rate, the researchers have limited their considerations to linear stress gradient, completely ignoring the time evolution of the stressed state. In the present paper we have made an attempt to combine these two directions. We present a model, reasonable from the viewpoint of physics, considering stress evolution during EM. Starting from stress evolution, drift kinetics is studied.

2. Model Since the defects responsible for EM phenomena in metals are vacancies, it would be most natural to study the evolution of vacancy structure of the conductor under the conditions when electric current *Corresponding author. 0167-9317 / 00 / $ – see front matter PII: S0167-9317( 99 )00301-9

 2000 Elsevier Science B.V. All rights reserved.

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passes through it. Let us consider the scheme in which the direction of electric current in the conductor of length l is opposite to the direction of the X-axis. There are three main factors defining stress and vacancy system evolution during EM. 1. EM flux of vacancies is determined by [1]: D JEM 5 C ] r jeZ 5VC kT

(1)

where C is relative vacancy concentration, D is the vacancy diffusion coefficient, r is resistivity, j is current density, e is electric charge, Z is efficient charge, V is EM rate. 2. Deviation from the initial equilibrium due to EM flux initiates the mechanism of inner sources and sinks of vacancies (lattice, grain boundary and interphase dislocations). It is accepted that the oxide film (Al 2 O 3 ) or passivation substantially decreases the ability of the surface to emerge or absorb vacancies [2]. The efficiency of this mechanism is so high in the majority of cases that the vacancies can be considered in local equilibrium with sources and sinks. The action of these sources and sinks in a body with fixed boundaries causes the appearance of stress due to changes in the lattice sites concentration [3]: ds dCat 5 C 0at dC 5 2 ] E

(2)

where dCat is a change in the lattice site concentration; E is Young’s modulus, C at0 is the equilibrium atomic concentration of vacancies in the conductor in the absence of current and mechanical stresses. It should be noted that the changes of mechanical stresses lead to the changes in equilibrium vacancy concentration: C 5 exp(sV /kT ),

(3)

where V is the volume of a vacancy. 3. This dependence is a consequence of the above-mentioned local equilibrium. The chemical potential of the vacancy can be presented as

m 5 kT ln C 1 sd V,

(4)

Here d V 5 va 2 V is the relaxation volume of a vacancy (va is the volume of an atom). The energy spent during the formation of a vacancy can be described by the equation: W 5 s va .

(5)

Taking (4) equal to (5) it is easy to obtain (3). 4. Finally, the last factor is the diffusion of vacancies connected with non-uniform distribution of chemical potential. The diffusion vacancy flux is D ≠m JD 5 2 ] C ] kT ≠x

(6)

Substituting (4) into (6) we obtain: ≠C JD 5 2 b D ] ≠x

(7)

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where b 5 va /V. So, to sum up the factors described, we obtain: ≠C ≠ 1 ≠s ] 5 2 ]s JD 1 JEMd 2 ]] ] 0 ≠t ≠x C at E ≠t

(8)

Using the Eq. (3) to transform the last term in the right-hand side of the equation and designating a 5 kT /(C 0at EV ), one can obtain the final equation a ≠C ] ] 5 b DCxx 2VCx (9) C ≠t in which the fact is taken into account that a 41 for the majority of important cases. The equation thus obtained differs only slightly from that obtained in Ref. [3]. Unlike Ref. [3], in the present work we pay major attention to the evolution of stresses under drift conditions. We shall focus our further considerations mainly on stationary solutions of this equation for different assumptions concerning boundary conditions. 3. Solutions and discussions

3.1. Stress distribution in pre-critical region ( jl),( jl)cr In the case of stationary conditions with zero vacancies flux, we obtain

b DCx 5VC

(10)

Let us designate r 5Vl /b D. The approximate solution of this problem for exp(r / 2) /r 2 < a (e.g., r,30 when a |10 5 ) is

S

V(x 2 l / 2) C 5 exp ]]]] bD

D

sV V(x 2 l / 2) ]] 5 ]]]] kT bD

(11) (12)

This solution coincides with that obtained earlier in Ref. [3] and corresponds to a symmetric distribution of stresses over the conductor length.

3.2. Critical length of the conductor Let us perform an analysis of the results obtained, in order to check whether the stresses arising in the conductor when current passes through it reach the yield limit. We shall consider the general case when yield limits for tension and compression are different. To reach yield limit at both ends of the conductor is to fulfil the conditions: C(0) 5 C 2 5 exp(s y2 V /kT )

(13)

C(l) 5 C 1 5 exp(s y1 V /kT )

(14)

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The common form of the distribution C(x) corresponding to zero flux is

S D

Vx C 5 A exp ]] bD

(15)

To meet (13), (14) and (15) simultaneously it is necessary that

S D

Vl exp ]] bD

C1 5] C2 cr

(16)

Thus, we can obtain the equation for the critical relation vass y1 2 s y2d va Dsy 5 ]] sjld cr 5 ]]]] Zer Zer

(17)

3.3. Drift conditions ( jl).( jl)cr Once ( jl) cr is exceeded, the boundary conditions (13) and (14) should be met. The given conditions already do not provide zero flux of vacancies at the boundaries. This flux is fully provided by plastic deformation and is connected with action of the same sources and sinks but in another manner, i.e., sliding of lattice dislocations and slipping over grain boundaries. In this case, the stationary solution is as follows:

S D

Vx C 5 A 1 B exp ]] bD

(18)

From (13) and (14) it follows that C1 2 C2 B 5 ]]] er 2 1

(19)

C 2e r 2 C 1 ]]]] A5C 2B5 er 2 1

(20)

2

Stress distribution under drift conditions is determined by the equation:

S

S DD

s (x)V Vx ]] 5 ln A 1 B exp ]] kT bD

(21)

The distribution becomes non-linear. Total vacancies flux for this case is Jv 5VC 2 b DCx 5VA

(22)

The rate of cathode end drift will be equal to: Vd 5VAC 0at

(23)

The dependence of stationary drift rate on conductor length is determined by the Eqs. (20) and (23). This dependence is presented in Fig. 1 in dimensionless form. As one can see, when the length of the line approaches the critical l cr , drift rate becomes dependent linearly on inverse line length, which was observed in the classical works of Blech [1].

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Fig. 1. Dimensionless form of dependence of the stationary drift rate on the length of the conductor (T5500 K, r 55310 26 V cm, Z55). Designation is accepted Vmax 5VC 0at C 2 .

3.4. Semi-infinite conductor Let us consider a semi-infinite conductor with the edge at x50. The stationary state in this problem meets the conditions C(0) 5 C 1 , C(2`) 5 1 The solution is as follows: Vx C 5 1 1 (C 1 2 1)exp ]] bD

S D

(24)

So, the vacancies flux is equal to Jv 5V and the rate of cathode end drift is Vd 5VC at0 .

(25)

3.5. Drift kinetics in a finite conductor Now we shall consider a conductor of finite size for which ( jl).( jl) cr . The evolution of stressed state and drift kinetics are presented in Fig. 2 and 3. This is the result obtained by solving Eq. (9) numerically. Drift kinetics exhibits three characteristic periods: 1. Induction period (Fig. 3). During the induction period, the stresses near the ends of the conductor increase (Fig. 2) until reaching yield limit. An approximate time of the induction period is 1 1 ab D (C 2 1) ln C t i 5 ]] 2 ]]]]] C1 V

(26)

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Fig. 2. Typical stress distributions for three moments of a time corresponding to the characteristic parts of the drift kinetics: (1) induction period; (2) quasi-stationary drift; and (3) stationary drift.

The time of induction period is inversely proportional to current density to the second power. 2. Quasi-stationary state. Stress distribution near the cathode end of the conductor is similar to the distribution in semi-infinite conductor. The stressed region spreads with time from the anode end over the whole length of the conductor (Fig. 2). Maximum drift rate in this case is equal to the

Fig. 3. Drift kinetics. Calculated dependence of the drift rate on a time (T5500 K, r 55310 26 V cm, Z55, D510 27 1 cm 2 / s, j510 6 A / cm 2 , l50.01 cm, s 2 y 5 s y 5200 MPa). The induction period is shown in the insert. Designation is 0 accepted Vat 5VC at

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stationary drift rate in semi-infinite conductor Vd 5VC at0 . Characteristic quasi-stationary drift duration is al t qs 5 ] (27) V 3. Stationary state. A steady distribution of stresses is established. It is determined by Eq. (21) (Fig. 2). Stationary drift rate is described by (23). 4. Conclusions 1. A necessary condition for drift is the achievement of yield limits at both the cathode and anode ends of the conductor. The critical relation ( jl) cr or the critical length of the conductor are determined by the difference in yield limits at the anode and cathode ends of the conductor. 2. In the over-critical region ( jl).( jl) cr , drift kinetics is determined by the evolution of stressed state in the conductor and can be described by three characteristic periods: a) induction period which is the time within which the stress increases until the yield limit is reached at the cathode end of the conductor; b) In the quasi-stationary state, the stressed region spreads from the anode end over the whole length of the conductor. Maximum drift rate is equal to the drift rate in semi-infinite Vd 5 C 0atV. With increasing conductor length, the time of quasi-stationary state increases; and c) stationary state is characterised by the steady distribution of stresses. Maximum stationary drift rate is determined by the yield limit at the anode end of the conductor Vd 5VC at0 exp(s y2 V /kT ). References [1] I.A. Blech, Electromigration in thin aluminium films on titanium nitride, J. Appl. Phys. 47 (1976) 1203–1208. [2] C.H. Hu, K.P. Rodbell, T.D. Sullivan, K.Y. Lee, D.P. Bouldin, Electromigration and stress-induced voiding in fine Al and Al-alloy thin-film lines, IBM J. Res. Dev. 39 (1995) 465–496. [3] M.A. Korhonen, P. Borgesen, Stress evolution due to electromigration in confine metal lines, J. Appl. Phys. 73 (1993) 3790–3799.