Influence of electric current on diffusionally accommodated sliding at hetero-interfaces

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Acta Materialia 59 (2011) 2096–2108 www.elsevier.com/locate/actamat

Influence of electric current on diffusionally accommodated sliding at hetero-interfaces P. Kumar, I. Dutta ⇑ School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA Received 29 June 2010; received in revised form 29 November 2010; accepted 5 December 2010 Available online 10 January 2011

Abstract Diffusionally accommodated sliding at metal/non-metal interfaces has been experimentally and analytically investigated under a combination of an interfacial shear stress and a superimposed electric current. It is demonstrated that interfaces slide faster if electron flow is in the direction of applied shear on the metal side of the interface, which causes the stress-driven diffusional flux flowing along the interface to be augmented by the interfacial flux due to electromigration (EM). Conversely, the sliding rate decreases if electrons flow opposite to the shear direction on the metal side, which causes the stress- and electromigration-induced fluxes to counteract each other. The contribution of electric current to sliding is significant only when a sufficient EM flux is associated with the interface, a situation which is common in modern electronic devices. An analytically derived constitutive model describing the kinetics of EM-influenced interfacial sliding is proposed. Consistent with experiments, the model predicts that the sliding kinetics have linear dependencies on stress and current density, and an Arrhenius dependence on temperature, with an activation energy equal to that for interfacial diffusion when the grain size is large and the temperature is relatively low. At small grain sizes, both interfacial and film diffusivities contribute to sliding, and the constitutive behavior becomes more complex. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Interfacial sliding; Diffusion; Stress; Electromigration; Current density

1. Introduction The elevated-temperature mechanical properties of interfaces between dissimilar materials are critical to the performance of a wide range of microsystems such as microelectronic and microelectromechanical devices. In most such applications, a metallic component, typically in the form of a thin film line, is in contact with a semiconductor or ceramic component, and the interface is subjected to large shear stresses. In addition, the metallic component often carries a large electric current density, leading to Joule heating of the entire system to high homologous temperatures (T/Tm). Together, the stress and high T/Tm enable diffusionally accommodated sliding at the

⇑ Corresponding author. Tel.: +1 509 335 8354; fax: +1 509 335 4662.

E-mail address: [email protected] (I. Dutta).

interface (interfacial creep) (e.g. [1–24]), impacting the reliability of the component. In addition, the high current density can enable electromigration (EM)-induced mass transport through the bulk (via volume or grain boundary diffusion in the metal) as well as along the hetero-interface [25–27], potentially leading to interaction between stress and EM influencing the kinetics of interfacial sliding. Stress-driven interfacial sliding has been studied in various model systems [3,16–18], and the key phenomenological aspects of a kinetics law [3] have been experimentally validated [16,17]. Interfacial sliding is driven by applied shear stresses at high T/Tm, where diffusion processes are rapid. Sliding occurs faster for smoother interfaces, and when the interface is stressed normally in tension in addition to shear [3,17]. The basic mechanism is akin to that of grain boundary sliding [28], where applied far-field stresses cause normal stress gradients at the interface due to topographical variations, allowing mass transport, and

1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.12.011

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hence sliding, along the interfaces. The interfacial shear displacement rate due to sliding under stress only is given by [3,17]: "  3 # C  di D i X 3 h _ U¼ si þ 2p rn ; ð1Þ k kTh2 where C is a numerical constant; di and Di are the interfacial thickness and diffusivity, respectively; X is the atomic volume of the diffusing species; k and h are the topographical periodicity and roughness of the interface, respectively; and k, R and T are Boltzmann constant, gas constant and temperature, respectively. The resulting interfacial shear strain rate is given by c_ i ¼ U_ =h, where the interfacial roughness h is the effective width of the interfacial region [17]. The interfacial shear stress si is the primary driving force for this process, which is enhanced if a far-field tensile normal interfacial stress rn is present (i.e. positive rn), or reduced if there is a compressive rn (negative rn). The effect of rn on sliding is finite only when si is non-zero, and rn = 0 when si = 0. Since U_ / h2 , smoother interfaces slide more rapidly. Stress-driven interfacial sliding has been noted during thermal cycling of metal-matrix composites [4,5], thin film–substrate systems [19–22], and metal–dielectric interconnect structures in microelectronic devices [21–24]. As microelectronic and microelectromechanical devices shrink in scale and the interfacial area per unit component volume increases commensurately, the importance of interfacial sliding in device reliability is expected to rise. In addition, the huge increase in interfacial area makes interfaces increasingly important paths for mass transport via EM [26,27]. For instance, it is now well known that in dual-damascene structures of Cu interconnect lines embedded within dielectrics in microelectronic devices, the interfaces between Cu and Si3N4 capping layers serve as the primary path for EM flux (JEM), rather than the grain boundaries within the Cu itself [26]. Since interfacial diffusion also drives interfacial sliding, diffusional fluxes due to EM and applied stresses may be expected to augment or mitigate each other, thereby influencing the kinetics of interfacial sliding. It is therefore important to develop a mechanistic understanding of the interplay between stress and electric currents in driving interfacial sliding in thin film structures. In this paper, we report on experiments on interfacial sliding based on a model metal–semiconductor system under the combined impetus of an interfacial shear stress and an electric current flowing through the metal, and present an analytical model to describe this interaction. 2. Experimental approach Si–Pb–Si sandwich structures were fabricated by diffusion-bonding a 10 lm thick foil of Pb between two 35 mm long, 6 mm wide Si strips cut from 450 lm thick polished undoped h1 0 0i Si wafers with an RMS roughness

Fig. 1. Microstructure of a Pb foil following rolling and annealing at 300 °C for 10 h.

of 10 nm.1 Prior to diffusion-bonding, the Si wafers were degreased, cleaned and deoxidized following standard microelectronics industry practice. The Pb foils were produced by rolling 99.999% pure Pb billets, and annealing at 300 °C for 10 h. The resulting grain size in the Pb foil, determined by the mean linear intercept method, was close to 150 lm, as shown in Fig. 1. The Pb grain size was chosen to be deliberately large in order to reduce the contribution of grain boundary diffusion within the bulk film, and commensurately enhance the relative contribution of diffusion along the Pb–Si interface to the EM flux. The Pb foils were cut into 3 mm wide strips, lightly etched in 15 vol.% HNO3 solution, and diffusion-bonded between two Si strips to form a lap-joint with an overlap length of 10 mm. Diffusion-bonding was performed at a temperature of 300 °C under an applied pressure of 120–130 MPa for 6 h at a vacuum of 8  105 torr, following which the sample was cooled very slowly to ambient temperature (at about 0.8 °C min1) in order to minimize thermal residual stresses ˚ thick within the Pb foil. For most of the samples, a 100 A Cr film was deposited by thermal evaporation on one of the Si strips on one side of the Pb foil prior to diffusionbonding. This produced lap-shear samples with a Pb–Si interface on one side, and a Pb–Cr–Si interface on the other, each interface being 10 mm  3 mm in area. Pb is nominally immiscible in the solid state with both Si and Cr [29]. The Pb–Si interfacial intermixed zone is of the order of a few angstroms [30–32], allowing the Pb–Si interfaces in the lap-shear joint to be relatively sharp. Cr also adheres strongly to Si by reacting with native surface oxides and well as by forming silicides (10 monolayers thick) following annealing at 300–400 °C [32–35], while forming a strongly adherent interface with a limited diffusion zone with Pb [36]. Therefore, the Pb–Cr–Si interface is expected to be resistant to diffusionally accommodated sliding,

1 The interfacial roughness h is nominally equal to the surface roughness of the Si wafer prior to diffusion-bonding.

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Fig. 2. (a). Schematic of the experimental arrangement for measuring interfacial creep. (b). Details of the lap-shear sample along with both mechanical and electrical loading configurations.

whereas the Pb–Si interface without the Cr interlayer serves as a short-circuit diffusion path and therefore slides readily. That this assumption is valid will be demonstrated experimentally. Aluminum tabs of appropriate thicknesses were attached to the ends of the silicon strips using a ceramic glue with a glass transition temperature of 350 °C in order to enable the samples to be loaded in shear using friction-fixtures for tensile testing, and to ensure coincidence of the test interface along the tensile axis. The entire experimental arrangement is shown schematically in Fig. 2a, and details of the sample along with both mechanical and electrical loading configurations are shown in Fig. 2b. In order to enable EM by passage of electric current through the Pb foil during shear-loading, two thin Cu wires were soldered to the foil protruding from the lap-joint, about 1–2 mm away from each end of the joint. A flexible Kapton foil-heater was attached on one side of the lap-joint, and a thin, flexible thermocouple was attached to the other side, allowing heating of the joint to test temperatures up to 150 °C. A capacitance displacement gauge (0.01 lm resolution) was affixed to the upper friction grip, and a steel plate attached to the lower grip served as the target for the capacitance gauge. The measured displacement between the upper and lower grips represent the sum of the shear displacements at the Pb–Si interface (dPb–Si) and the Pb–Cr–Si interface (dPb–Cr–Si  0), and of the Pb foil (dPb), i.e. dmeas = dPb–Si + dPb–Cr–Si + dPb  dPb–Si + dPb. It was ascertained that the shear displacement of the Pb foil under all test conditions was negligible (i.e. dPb–Si  dPb  0),

yielding dmeas  dPb–Si.2 During each test, a constant tensile load was applied to the sample by hanging dead weights from the lower grip (Fig. 2a), and the load was recorded using a load cell attached to the upper pull-rod. Since the interfacial displacements were negligible in proportion to the length of the lap-shear joint (i.e. the interface), constant applied loads produced a nominally constant shear stress (s) along the entire length of the interface. The entire assembly was housed inside an insulated environmental chamber filled with argon allowing a temperature stability of ±0.5 °C and a displacement stability of ±20 nm over the entire test period (up to 48 h). Following each test under applied stress only, a constant electric current was applied to the Pb film using a high current power source, while still maintaining the applied load. The electric current was first applied in the direction opposite to that of the shear force acting on the Pb side of the Pb–Si interface (negative current density, j), and, the current direction was then reversed (positive current density, j). A few experiments were also conducted without an applied force, using a modified experimental set-up3

2

That dmeas  dPb–Si and dPb  dPb–Si is demonstrated in Section 3.1. In the modified set-up, the sample was placed in a horizontal cantilever configuration. One of the Si strips of the lap-shear specimen was fixed in a grip to which the capacitance gauge was attached, while the other Si strip was unattached to any fixture and free to move. A small Al target attached directly to the free Si strip served as a reference surface for the capacitance gauge. 3

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to inspect the effect of electric current only on any displacement at the two interfaces (both Pb–Si). Fig. 3a and b, where the interface of interest (i.e. Pb–Si) is depicted to have periodic topographic undulations, show the directions and sign conventions of mechanical loading (s), electron flow (e, which is opposite to j), and the electric field (E = jq, where q = resistivity of Pb). The corresponding directions of the fluxes due to s and EM (Js and JEM respectively) are also shown. s is arbitrarily defined to be positive when the shear direction on the Pb side of the right-hand interface is in the +Y direction. j is also positive when it is in +Y direction. When a positive s is applied on the Pb–Si interface on the right-hand side, the Si on the right moves down (in the Y direction). This causes the Pb adjoining the right-hand interface to move in the +Y direction relative to the Si, and the corresponding interfacial displacement U is positive, as is Js. When a positive current density j is applied to the Pb foil, the current flows in the +Y direction, and the corresponding electron flow (e) is in the Y direction, leading to a negative JEM, which counteracts Js (Fig. 3b). Conversely, when j is negative, electron flow is in the +Y direction, and JEM acts in tandem with Js (Fig. 3a). One edge of a few of the samples was metallographically polished to 0.05 lm, and a thin film Al grid pattern of 7.5 lm  7.5 lm squares was thermally evaporated on to this surface for to enable visual observation of the extent of interfacial sliding displacement following testing.

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3. Results 3.1. Observation and measurement of interfacial creep Fig. 4a and b show displacement vs. time plots (U vs. t) obtained from the interfacial creep tests at two different applied shear stresses (s = 0.5 and 0.17 MPa, respectively), with and without applied j. In all experiments, an apparent steady-state creep behavior was established in well under 1 h. At small s, the data are noisy since the displacements are small, but still show a nominally constant displacement rate U_ during the test. Importantly, it is noted that when a negative j is superimposed on the applied stress (i.e. when the direction of electron flow in the Pb foil is the same as the shear direction on the Pb side of the Pb– Si interface, as in Fig. 3a), U_ increases, whereas when j is positive (i.e. the electron flow direction is opposite to the shear direction on the Pb side of the Pb–Si interface, as in Fig. 3b), U_ decreases. At large values of s, where the displacement rate due to stress only is relatively large, a positive j simply decreases the sliding rate U_ , while a negative j increases U_ (Fig. 4a). On the other hand, when the sliding rate due to stress only is small (small U_ due to small s), the application of a positive j may reverse the displacement direction, making U and U_ negative (Fig. 4b). These observations clearly establish that the sliding kinetics and direction are determined by the combination of applied s and j, thus suggesting an interaction between stress-assisted diffusion and EM in inducing interfacial sliding. A positive

Fig. 3. Schematics showing the directions and sign conventions of mechanical loading (s), electron flow (e, which is opposite to j), electric field (E), and the fluxes due to s and EM (Js and JEM respectively). The interface of interest (Pb–Si) is depicted to have a periodic topography.

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Fig. 4. Displacement vs. time plots (U vs. t) obtained from the interfacial creep tests at two different applied shear stresses (s = 0.5 and 0.17 MPa), with and without applied j.

s and negative j give interfacial diffusional fluxes Js and JEM which are additive, whereas a positive s and positive j yield Js and JEM acting in opposite directions. Any EM-induced mass transport necessarily occurs in the same direction along both interfaces adjoining the Pb foil (i.e. JEM is always in the direction of electron flow, which is the same on the Pb side of both interfaces). On the other hand, diffusional flow due to the applied shear stress occurs in opposite directions along the Pb sides of the two interfaces (Js along each interface is along the directions of s on the Pb side, which are opposite to each other, as elucidated in Fig. 3). Therefore, the impact of EM on sliding would be undetectable if both interfaces were able to slide. Because of this, it was critical for these experiments to have one non-sliding interface (i.e. the Pb–Cr–Si interface on one side of the Pb foil). Fig. 5a and b elucidate the role of the Cr interlayer in dramatically reducing the interfacial sliding rate. Fig. 5a shows displacement vs. time plots under s only, as well as due to a combination of s and j (negative), for two types of samples—with a Cr layer on one side of the lead foil,

Fig. 5. (a). Displacement vs. time plots under s only, and both s and j (negative), for samples with a Cr layer at one or both interfaces. (b). Scanning electron micrograph of a sample with fiduciary grids deposited on a polished edge, following a test with s = 0.5 MPa and j = 2.36  104 A cm2.

and with a Cr layer at both interfaces. Clearly, when Cr is present at both interfaces, the overall sliding rate U_ is an order of magnitude smaller than when only one of the interfaces has Cr, under the same s and j conditions. This suggests that the Cr interlayer is effective in substantially slowing sliding. Fig. 5b shows a scanning electron micrograph of a sample with Al thin film fiduciary grids deposited on a polished edge, following a test with s = 0.5 MPa and j = 2.36  104 A cm2. The sample, which has been tilted relative to the electron beam to emphasize the Al grid pattern, clearly demonstrates that all deformation was confined to the Pb– Si interface only, with negligible deformation at the Pb– Cr–Si interface, or in Si or Pb. The observed displacement was also uniform along the entire interface, and consistent with the total measured displacement. No evidence of interfacial decohesion was noted. This establishes conclusively that the displacements measured during a test were largely due to sliding at the Pb–Si interface (i.e. dmeas  dPbSi i/f). A comparison of the measured displacement rate U_ with computed displacement rates due to shear creep of the Pb foil or due to EM also showed that U_ is several orders of magnitude larger than those due to either phenomena alone.

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(At 354.5 K, the measured U_  3:5  102 nm s1 at s = 0.5 MPa and j = 2.36  104 A cm2, whereas the computed U_ value due to combined diffusional and climbcontrolled creep of Pb with 150 lm grains at s = 0.5 MPa is 1.43  107 nm s1, and that due to EM via both bulk and grain boundary diffusion in Pb of 150 lm grain size at j = 2.36  104 A cm2 is 9.78  107 nm s1.) Therefore, we infer that by far the predominant component of the measured displacement is that due to diffusionally accommodated interfacial sliding, i.e. U  Ui. 3.2. Interfacial sliding due to electric current only Fig. 6a shows the displacement vs. time plot obtained from a test conducted at j = 2.43  104 A cm2 and si = 0, at a temperature of 108.5 °C. Several micrometers of relative displacement are observed between the Si strips in the sandwich sample. Fig. 6b and c shows scanning electron micrographs of the edge of a sample before and after application of electric current for 72 h. It is clear from the deformation of the Al thin film fiduciary grid pattern that the measured displacement is largely confined to the two Pb–Si interfaces, with negligible deformation within the Pb foil itself. As such, the behavior is identical to that when

Fig. 6. (a) Displacement vs. time plot obtained from a test conducted at j = 2.43  104 A cm2 and si = 0, at a temperature of 108.5 °C. Scanning electron micrographs of the sample with fiduciary grids deposited on a polished edge before (b) and after (c) the test. Arrows indicate locations where evidence of sliding may be observed.

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both si and j are applied (Fig. 5b). This demonstrates that as long as the EM flux flows primarily through the interface (a situation that prevails when the grain size is large, and the interface is a rapid diffusion path), interfacial sliding can occur due to EM only (i.e. in the absence of an applied shear stress). The mass flow along the interface is along the direction of electron flow, and as a result, the substrate moves relative to the film in the opposite direction to electron flow. Thus, in essence, the applied electric field applies a periodically varying “electrical stress” along the interface, which, similar to the situation under an applied mechanical shear stress, allows mass transport along the interface to cause relative motion between the film and substrate, even when there is little mass flow through the film. 3.3. Constitutive behavior Fig. 7a and b show the effect of applied j on the U_ vs. s behavior at a constant temperature, and the effect of s on U_

Fig. 7. Plots showing (a) the effect of applied j on U_ vs. s at a constant temperature, and (b) the effect of s on U_ vs. j at constant temperature.

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66 kJ mol1, respectively), the much faster interfacial diffusion (Qi  48 kJ mol1) clearly dominates, allowing Eq. (2) to be rewritten as: Q

Q

i i U_ ¼ C 3 seRT  C 4 EeRT ;

ð3Þ

where C3 and C4 are constants (both positive) and E is the applied electric field (E / j). In the following, we present a model elucidating the theoretical basis of Eq. (3), and discuss the interaction between stress and EM in driving interfacial sliding. 3.4. Model for electromigration-aided interfacial sliding

Fig. 8. Temperature dependence of U_ at a constant s value, with and without an applied j. Under both conditions, U_ shows an Arrhenius temperature dependence, with an activation energy Q of 47–49 kJ mol1.

vs. j behavior at constant temperature, respectively.4 At all values of j (zero, negative or positive), the U_ vs. s plot is linear (Fig. 7a), suggesting that interfacial sliding occurs via a viscous flow mechanism, in agreement with Eq. (1). Furthermore, the U_ vs. j plots are also linear (i.e. U_ / j) at all applied s (Fig. 7b). Additionally, it is noted that the application of j raises or lowers the U_ vs. s plot by similar extents at all s values, while the application of s raises or lowers the U_ vs. j plot by similar amounts at different j values.5 This suggests that as a first approximation, the impacts of j and s on U_ may be considered to be nominally independent of each other, allowing us to represent U_ as: U_ ¼ C 1 s  C 2 j;

ð2Þ

where the terms C1 and C2 incorporate material properties and are nominally constant at a given temperature. Fig. 8 shows the temperature dependence of U_ at a constant s value, with and without an applied j. Under both conditions, ln U_ shows a linear dependence on 1/RT with the same nominal slope, indicating that both C1 and C2 show Arrhenius temperature dependencies, with an activation energy Q of 47–49 kJ mol1. This Q value is half that of volume diffusion in Pb, and is close to the activation energy of 50 kJ mol1 noted for diffusion along Pb–SiO2 interfaces [3]. This suggests that the sliding observed along the Pb–Si interface in the present study is controlled by interfacial diffusion, with the driving forces being provided by the applied shear stress and EM (which occurs opposite to the direction of j), the effects of these being additive. Although both stressand EM-driven sliding can have contributions from volume and grain boundary diffusion (activation energies of 104 and 4

In these experiments, there is no normal stress on the interface (i.e. rN = 0). 5 Although the slopes of the plots in Fig. 7a or b are not identical, the differences are small, and therefore can be assumed to be nominally invariant within the uncertainties in the experimental data.

The constitutive model developed here considers a metallic film attached to a semiconductor (or insulator) substrate which is subjected to an in-plane shear stress s, as well as an electric current density j which flows through the film, as shown in Fig. 9a. The model is similar to that developed by Raj and Ashby [28] for grain boundary sliding, later adapted by Funn and Dutta [3] for diffusionally accommodated interfacial sliding driven by a shear stress, but incorporates the impact of EM in the film. Based on the analysis approach of Raj and Ashby [28], the interface is considered to have a periodic topography with a cosine dependence on position (Fig. 9a):   h 2p x ¼ cos y ; ð4Þ 2 k where h is twice the amplitude of the interface, k is the interfacial periodicity, and x and y are the coordinates normal to, and along the interface, respectively. When an in-plane shear stress s is applied to the system, an instantaneous elastic deformation occurs at the interface, resulting in a periodically varying normal stress rn along the interface, as shown in Fig. 9b [3]:   2sk 2p sin y : ð5Þ rn ¼  ph k The variations in rn set-up chemical potential gradients along the interface, and lead to stress-assisted diffusion with the flux given by: D ~ ~ rðrXÞ; J stress ¼ XkT

ð6Þ

where D is the effective diffusivity through the film and/or interface. The electric current density j flowing through the film from the anode to the cathode (right to left in Fig. 9a) causes an EM flux JEM in the opposite direction, given by: D ~ D ~ ~ Z eE ¼  Z erV ; J EM ¼ XkT XkT

ð7Þ

where Z is the effective charge, e is the charge of an electron (Z e is a negative number), and ~ E (=the gradient of ~ ) is the applied electric potential V withinthefilm ¼ rV field. The field is related to the applied current density j by E = jq, where q is the resistivity of the metal film.

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Fig. 9. (a). Schematic showing an interface with periodic topography between a metallic film and a substrate, which is subjected to an in-plane shear stress s as well as an electric current density j which flows through the film. The curved arrows adjacent to the interface show the interfacial electric field vectors. (b). The far-field s results in a periodically varying normal stress on the interface (rn). (c) Magnitude of the electric field E acting tangentially along the periodic interface.

When an electric current (or field) is applied to a conductor with internal physical perturbations (e.g. an elliptical hole), the current density vectors (and electric field vectors) conform directionally to, but vary in magnitude along, the boundary of the perturbation [37]. Therefore, at a periodic interface between a conductor and an insulator, which may be thought of as a conductor surface with a series of perturbations, the electric field would be tangential to the interface and vary periodically, being concentrated at the “troughs” in the film (y = nk/2 for odd n) and weakest at the “hills” (y = mk/2 for even m), as shown in Fig. 9c. This variation becomes stronger with increasing aspect ratio of the perturbation (in this case, h/k), but with a commensurate decrease in the field strength at the hills [37]. For shallow perturbations (i.e. relatively smooth interface with h/k  1), this attenuation is negligible, and the field strength at the hills is nominally identical to the far-field applied E. This leads to the following boundary conditions for the Laplacian $2V(x, y) = 0, which represents charge conservation within the film: (1) oV/oy = E at x = 0 (interface) and y = mk/2 (m = even); (2) oV/oy = E at x = 1 (bulk) at all y; and (3) oV/ox = 0 at x = 0 (interface) and y = mk/2 (m = even). This yields, for the potential V at any location (x, y) within the film:    k 2px 2p V ðx; yÞ ¼ E ðk  yÞ  e k sin y ; ð8Þ p k which produces a periodically varying potential at the interface (x = 0):    k 2p V ð0; yÞ ¼ E ðk  yÞ  sin y : ð9Þ p k

Conservation of mass within the film (i.e. for all y and 0 6 x 6 1) requires that: r2 lðx; yÞ ¼ 0;

ð11Þ

where l, the chemical potential, is l(x, y) = l0  Xrn (x, y)  Z eV(x, y). Assuming that mass is conserved within each film length-segment equaling one wavelength of the interface (i.e. y = 0 to k), with the boundary conditions: lð0; yÞ ¼ l0  Xrn ð0; yÞ  Z eV ð0; yÞ

lð1; yÞ ¼ l0  Z eV ð1; yÞ

ð12aÞ ð12bÞ

one obtains: lðx; yÞ ¼ l0  Z eEðk  yÞ     2kX h 2p 2p si þ Z eE y : ð13Þ þ e k x sin ph 2X k Maintenance of continuity across the interface requires that the flux of matter into or away from the interface over any time increment accounts for the local displacement of the boundary along the y-direction. This yields, for the average interfacial displacement rate U_ :   @J y U_ sin h cos h ¼ X J xf ð0; yÞ cos h þ di i ; ð14Þ @y where di is the thickness of the interface, and J xf and J yi are the fluxes through the film and the interface, respectively. This treatment is identical to that in Ref. [3]. Assuming that the interface is relatively smooth, i.e. h/k  1 and h ? 0, Eq. (14) may be rewritten as: J xf ð0; yÞ  di

@J yi ð0; yÞ dx U_ : ¼ @y dy X

ð15Þ

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stress, whereas the second term represents the effect of the applied electric field E. The effective film diffusivity Deff f incorporates the effects of both volume and grain boundary diffusion, and may be expressed as: Deff ¼ Dvol þ f

pdgb Dgb /; d

ð17Þ

where Dvol is the film volume diffusivity, Dgb is the film grain boundary diffusivity, dgb is the grain boundary thickness, and / is a constant which is a function of d/k, d being the film grain size. In Eq. (17), / is defined such that / ? 0 when d > k. Since the interface periodicity k for most film– substrate interfaces is generally much smaller than d, Deff  Dvol in most cases. Also, since the activation energy f for interfacial diffusion Qi is typically much lower than that for volume diffusion Qvol (as noted in Fig. 8 and in Refs. [3,17]), it can be assumed that, in most cases, Di  Deff f . When Di  Deff f , the flux through the film is negligible relative to that along the interface, and Eq. (16) can be simplified to represent the sliding rate as: 8Xdi Di 4di Di U_ ¼ Z eE: si þ kTh kTh2

ð18Þ

Here the first term is the sliding displacement rate under applied shear when all flux is along the interface, whereas the second term represents the interfacial sliding rate due to EM only when interfacial diffusion dominates. Eq. (18) is identical in form to Eq. (3), recognizing that the exponential term in Eq. (3) is incorporated in Di ð¼ Q=RT Dio  ei Þ and that Z e is negative. Thus the constants in Eq. (3) are given by: C3 ¼

Fig. 10. Computed plots of U_ as a function of the magnitude of j (jjj, where j;= E/q) for (a) Pb–Si and (b) Al–Si interfaces at constant temperature.

eff

Df ~ Knowing that ~ J f ðx; yÞ ¼  XkT rlðx; yÞ and J yi ð0; yÞ ¼ Di @lð0;yÞ ~ ¼ @l þ @l, and combining , recognizing that rl  XkT @x @y @y these relationships with Eqs. (13) and (15), one may obtain a general equation for the interfacial sliding displacement rate ðU_ Þ when both EM and stress-driven diffusive fluxes may be transported along the interface as well as through the film (by volume and grain boundary diffusion). This gives:   4Xksi 2pdi Di eff _ U¼ Df þ k pkTh2   2kZ eE 2pdi Di Deff þ þ ; ð16Þ f pkTh k

and Di are the effective diffusivity through the where Deff f film, and through the interface, respectively. The first term on the right-hand side represents the effect of applied shear

8di Dio X kTh2

and

C4 ¼ 

4di Dio Z e: kTh

ð19Þ

The interaction between these two effects is shown in Fig. 10a and b, which plot U_ as a function of the magnitude of j (jjj, where j = E/q) for Pb–Si and Al–Si interfaces at constant temperature.6 Pb–Si represents the current experimental system, and Al–Si is an interface of technological relevance in microsystems applications. For both interfaces, it has been assumed that Di  Deff f , and that the roughness h is 10 nm. The trends observed for both interfaces are identical. Clearly, at low j values, the contribution of EM is negligible, and U_ is controlled principally by stress-driven sliding. However, with increasing j, the EM component results in an appreciable fraction of the total U_ . Importantly, the model results in the same behavior as those observed experimentally, with reversal in the current direction exerting opposite influences on the overall U_ . Since the term Z e in Eq. (18) is negative, a negative j (which leads to a negative E) results in a positive 6 Since the term diDio (the pre-exponent for interfacial diffusivity, diDi) is not known, it has been assumed to be: diDio = Ci. dgbDgbo, where Ci is a constant (108–106), and dgbDgbo is the pre-exponent for grain boundary diffusion coefficient [3,17]. The data for Al–Si interfaces have been obtained from Ref. [18]. The Z values for Pb and Al are 4 and 30, respectively [38].

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rate, and the dependence is linear, as noted in Fig. 7a. Secondly, the larger the jjj, the greater the influence of EM on the sliding rate, with a negative j augmenting U_ , and a positive j diminishing U_ . This trend is identical to that observed in Fig. 7b. Thirdly, with increasing temperature, both stress and j dependence of U_ become stronger, as reflected in the increase in the slopes of the U_ vs. s plots, as well as the greater separation between the plots for the same j values at higher temperatures. 4. Discussion

Fig. 11. Plots of the computed U_ vs. s behavior for different j values at different temperatures for (a) Pb–Si and (b) Al–Si interfaces.

EM-induced sliding rate, and acts additively to stress-induced sliding (when s is positive). Conversely, a positive j counteracts stress-induced sliding, reducing the total U_ . Furthermore, in agreement with experimental observations (Fig. 4b and b), it is apparent from Fig. 10a that when s is small and sliding due to s only is slow, a positive j (which results in an electron flow direction counteracting the shear direction on the film side of the interface) can make the sliding rate negative. Finally, Fig. 10b shows that when there is no applied stress (i.e. s = 0), the interface can still slide (under the influence of EM only), U_ being given by the second term in Eq. (18) (or Eq. (16) when Di and Deff are comparable). f Fig. 11a and b plot the computed U_ vs. s for different j values at different temperatures for the Pb–Si and Al–Si interfaces. The plots are consistent with the experimental behaviors shown in Fig. 7. Three trends are apparent in both systems. First, an increasing s increases the sliding

Under the application of stress only, interfacial sliding occurs via diffusional creep with a stress exponent of unity, its rate increasing with increasing interfacial roughness h [3,18]. This sliding occurs due a diffusive flux associated with the interface (Ji), the atomic flux being conducted not only by the actual interface, but also a region of the metallic component (in this case, the film or foil) immediately adjacent to the interface, with a thickness on the order of h. Since the flux arises due to normal stress gradients along the interface, little diffusional flux is carried through the bulk of the film itself (i.e. a distance significantly greater than h from the interface), and therefore, interfacial sliding does not necessarily result in deformation of the film. In this case, the film can simply slide as a nominally rigid body over the substrate. On the other hand, when a current is applied, a significant EM flux can be carried through the film itself if Deff f is high (i.e. at high temperatures and small grain sizes), leading to potential shape changes of the film (e.g. transport of matter from the tail end of the film to the leading edge, resulting in growth of protrusions [25,26]). However, when diffusional transport through the bulk of the film is limited due to a large grain size, the atomic flux due to EM is conducted through the interface [26,27], and this can lead to an equivalent situation as stress-driven sliding in that deformation or shape change of the film is not a prerequisite to sliding. This is the situation under which stress- and EM-driven sliding interact with each other, and is the condition under which the present experiments were conducted. Since Ji is carried by the interface as well as a thin region of the film, volume and grain boundary diffusion in the film next to the interface can influence interfacial sliding. The relative roles of volume, grain boundary and interfacial diffusion in determining the sliding kinetics are dependent on the temperature, and the ratio of grain size to interfacial periodicity (d/k). Since sliding occurs when atoms are transported by a distance equal to one period (i.e. k) of the interface, the role of grain boundary diffusion is negligible when d is greater than the diffusion distance (k). The smaller d becomes relative to k, the larger the contribution of grain boundary diffusion. Furthermore, as temperature increases, volume diffusion assumes enhanced importance, and hence diffusion through the film becomes increasingly important. These effects are illustrated in Fig. 12, which

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Fig. 12. Plots of interfacial sliding rate as given by Eq. (16) (which considers Dvol, Dgb and Di) and Eq. (18) (Di only) for the Pb–Si interface.

plots Eq. (16) (which considers Dvol, Dgb and Di) and (18) (Di only) for the Pb–Si interface. At low temperatures (353 K), if d > k (i.e. the contribution of grain boundary diffusion is insignificant), the sliding rate is independent of whether Eq. (16) or (18) is plotted. If, however, d 6 k, the sliding rate jumps dramatically because of grain boundary diffusion, as observed at 353 K when all diffusion components operate. With increase in temperature (e.g. 423 K), Dvol increases, and Eqs. (16) and (18) produce different sliding rates even when d > k. Therefore, while the simplified behavior represented by Eq. (18) may be used at low temperatures and when d > k, it may be necessary to use Eq. (16) in other cases. Writing the driving forces for stress-driven sliding and EM-driven sliding as: si X F stress ¼ ; F EM ¼ Z eE; ð20Þ k where the interfacial periodicity k the diffusion distance for stress-driven sliding for h  k, Eq. (18) becomes:   8di Di k 4di Di F EM : U_ ¼ ð21Þ F stress þ kTh h kTh For the Pb–Si interface, with s = 0.2 MPa and j = 2  104 A cm2, which are close to the experimental conditions, and assuming that h = 10 nm, Fstress  6.1  1017 N and FEM  4.1  1016 N. Thus, in this case, FEM is about an order of magnitude larger than Fstress. Assuming that k/h  10, the contributions of stress and EM to interfacial sliding, as given by Eq. (21), are thus roughly comparable. An inspection of Eqs. (18) or (21) reveals that the stress and EM contributions to U_ are proportional to 1/h2 and 1/ h, respectively. As a result, stress-driven sliding dominates at very small roughnesses (<1 nm), whereas EM-driven sliding is expected to dominate at large roughnesses (>100 nm). This is shown in Fig. 13a and b, which plot

Fig. 13. Computed plots of U_ vs. h for a given combination of s and j for (a) Pb–Si and (b) Al–Si interfaces, respectively.

U_ vs. h for a given combination of s and j for Pb–Si and Al–Si interfaces, respectively. In the intermediate regime (5–100 nm), which is relevant for most interfaces for microsystems applications where thin films are subjected to electric current as well as stress, both mechanisms play important roles, their relative contributions depending on the specific combination of s and j. In Eq. (18), since Z is negative, a negative E (which causes electron flow in the same direction as s) gives rise to stress-directed and EM fluxes in the same direction, which act additively to enhance the sliding rate. Conversely, when E is positive, Js and JEM act subtractively, reducing the sliding rate. This suggests that interfacial sliding, were it to become a problem in electronic interconnects, can be mitigated by designing circuits in such a way that the direction of shear on the metallic component is counteracted by the electron flow. This design methodology is expected to be particularly advantageous for nanoscale interconnects with bamboo grain structures

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operating at relatively low homologous temperatures, where the entire EM diffusive flux can be transported along the interface, thereby making it either highly susceptible to sliding (s and j with opposite signs), or highly resistant to sliding (s and j with the same sign). 5. Conclusions Experimental evidence has been provided that the kinetics of diffusionally accommodated sliding at metal/ non-metal interfaces, which is known to occur under a combination of shear and normal stresses acting on the interface, can be enhanced or mitigated by a superimposed electric current. The applied interfacial stresses set-up periodic normal stress gradients along the interface, which can drive an interfacial diffusional flux at elevated homologous temperatures. When the metal component also carries a sufficiently high current density causing an appreciable EM flux along the interface, the stress- and EM-driven fluxes interact. When the current direction through the metallic component is opposite to the shear direction on the metal side of the interface (i.e. the electron flow direction in the metal is the same as the shear direction), the two fluxes act additively and enhance the rate of sliding. Conversely, when the electron flow direction is opposite to the direction of shear on the metal side, the fluxes oppose each other and reduce the rate of sliding. The observed sliding occurs due to diffusional transport along the physical interface (interfacial diffusion), as well as through a thin region of the metal film immediately adjacent to the interface (grain boundary and volume diffusion). When the interfacial periodicity k, which represents the unit diffusion distance for sliding, is small compared to the grain size of the film, interfacial diffusion dominates. While the stress-driven flux is always associated with the interface, the EM flux can flow through the bulk as well as the interface. The contribution of electric current to sliding is significant only when a sufficient EM flux is associated with the interface, a situation which is common in modern electronic devices. An analytically derived constitutive model describing the kinetics of interfacial sliding due to the interaction between stress- and EM-driven fluxes has been developed. The trends predicted by the model are in good qualitative agreement with the experimental data. Consistent with the experiments, the model predicts that the sliding kinetics have a linear dependence on stress and current density, and an Arrhenius dependence on temperature, with an activation energy equal to that for interfacial diffusion when the grain size is large and the temperature is relatively low. At small grain sizes, both interfacial and film diffusivities contribute to sliding, and the constitutive behavior becomes more complex. Furthermore, in agreement with experiments, the model correctly predicts the augmentation or mitigation of the sliding rate due to a reversal of applied current for a fixed applied shear direction. The model also shows that the sliding rate increases

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with decreasing interfacial roughness h, the stress-driven component of the rate being proportional to 1/h2, and the EM-driven component being proportional to 1/h. As a result, the former dominates at very smooth interfaces, while the latter dominates at relatively rough interfaces, with both components producing comparable rates at intermediate roughness. Finally, the model showed that a normal interfacial stress enhances the driving force for stress-driven sliding if it is tensile, and reduces it if it is compressive, whereas the back-stress due to EM, which is always compressive, reduces the driving force for EMdriven sliding. The new phenomenon of EM-influenced interfacial sliding reported here is expected to be important in emerging electronic devices with nanoscale thin film features, which are subjected to far-field shear and normal stresses due to packaging, as well as very high applied current densities, and where the large grain size (relative to the feature size and diffusion distance) forces the diffusional flux to flow along the interfaces. The results presented here also suggest a possible way to mitigate interfacial sliding by designing the device circuitry in such a way that the electron flow direction opposes the shear direction in the metallic features. Acknowledgements This work was supported by the National Science Foundation, Division of Materials Research, under Grant No. DMR-0513874. The authors gratefully acknowledge the assistance of Prof. A. Gopinath and Dr. T. Chen (both of NPS) in checking the mathematical integrity of the model. References [1] Yoda S, Kurihara N, Wakashima K, Umekawa S. Metall Trans A 1978;9:1229. [2] Dutta I, Mitra S, Wiest AD. Some effects of thermal residual stresses on the strain response of graphite–aluminum composites during thermal cycling. In: Barrera EV, Dutta I, editors. Residual stresses in composites. Warrendale (PA): TMS-AIME; 1993. [3] Funn JV, Dutta I. Acta Mater 1999;47:149. [4] Nagarajan R, Dutta I, Funn JV, Esmele M. Mater Sci Eng A 1999;259:237. [5] Dutta I. Acta Mater 2000;48:1055. [6] Zhmurkin DV, Gross TS, Buchwalter LP. J Electron Mater 1997;26:791. [7] Jobin VC, Raj R, Phoenix SL. Acta Metall Mater 1992;40:2269. [8] Ankem S, Margolin H. Metall Trans A 1983;14:500. [9] Ankem S, Margolin H. Metall Trans A 1986;17:2209. [10] Rosler J, Evans AG. Mater Sci Eng A 1992;153:438. [11] Mishra RS, Bieler TR, Mukherjee AK. Acta Metall Mater 1995;43:877. [12] Mishra RS, Bieler TR, Mukherjee AK. Acta Mater 1997;45:561. [13] Ignat M, Bonnet R. Acta Metall 1983;31:1991. [14] Gupta D, Vieregge K, Gust W. Acta Mater 1999;47:5. [15] Hsiung LM, Nieh TG. Creep deformation of lamellar TiAl alloys controlled by viscous glide of interfacial dislocations. In: Baker I, Noebe RD, George EP, editors. Interstitial and substitutional solute effects in intermetallics. Warrendale (PA): TMS; 1998.

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