Eect of Electromigration on Onset of Morphological Instability of a Nanowire Mikhail Khenner1 2

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Eﬀect of Electromigration on Onset of Morphological Instability of a Nanowire

Mikhail Khenner∗1, 2

1Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA

2Applied Physics Institute, Western Kentucky University, Bowling Green, KY 42101, USA

(Dated: April 4, 2023)

Solid cylindrical nanowires are vulnerable to a Rayleigh-Plateau-type morphological instability.

The instability results in a wire breakup, followed by formation of a chain array of spherical nanopar-

ticles. In this paper, a base model of a morphological instability of a nanowire on a substrate in

the applied electric ﬁeld directed along a nanowire axis is considered. Exact analytical solution is

obtained for 90◦contact angle and, assuming axisymmetric perturbations, for a free-standing wire.

The latter solution extends the 1965 result by Nichols and Mullins without electromigration eﬀect

(F.A. Nichols and W.W. Mullins, “Surface-(Interface-) and volume-diﬀusion contributions to mor-

phological changes driven by capillarity”, Trans. Metall. Soc. AIME 233, 1840–1848 (1965)). For

general contact angles the neutral stability is determined numerically. It is shown that a stronger

applied electric ﬁeld (a stronger current) results in a larger instability growth rate and a decrease of

the most dangerous unstable wavelength; in experiment, the latter is expected to yield more dense

chain array of nanoparticles. Also it is noted that a wire crystallographic orientation on a substrate

has larger impact on stability in a stronger electric ﬁeld and that a simple switching of the polarity

of electrical contacts, i.e. the reversal of the direction of the applied electric ﬁeld, may suppress the

instability development and thus a wire breakup would be prevented. A critical value of the electric

ﬁeld that is required for such wire stabilization is obtained.

Keywords: Nanowires, morphological stability, electromigration

I. INTRODUCTION

Theoretical studies of morphological stability and evolution of single-crystal solid wires (aka cylinders, rods, or

whiskers) by surface diﬀusion have long history that begins with the 1965 work by Nichols and Mullins [1, 2]. These

authors considered linear stability of a free-standing wire with respect to axisymmetric perturbations and found [1]

the dimensionless perturbations growth rate σ=k2(1 −k2), which is a classical expression of the Rayleigh-Plateau

instability. In dimensional coordinates this implies that an axisymmetric perturbation induces instability if and only if

the wavelength is longer than the circumference of the undisturbed cylinder. More studies followed, where instability

was investigated in various increasingly complicated settings, such as surface energy anisotropy, stress, and contact

lines with the substrate [3–10]. These studies provided better understanding of the mechanisms responsible for wire

breakup [11–15]. Recent models and computations are focused on solid-state dewetting of islands of various shapes,

including wires [16–20].

Electromigration [21–24], on the other hand, has been modeled extensively as a simple and inexpensive technique

to bias surface diﬀusion and thus aﬀect and shape morphological and compositional instabilities and evolutions

[25–39]. Understanding surface electromigration-driven instabilities of wires is important for reliable and scalable

manufacturing of nanocontacts [40–42]. Applied multi-physics models of nanowire failure due to electromigration

were published [43], however, to our knowledge, there is no publications that theoretically address the fundamentals

of electromigration-driven wire instability and evolution.

In 1996, McCallum et al. [4] published the inﬂuential paper on stability of a wire with isotropic surface energy. In

their work, the wire is assumed deposited onto a substrate, with the wire surface making two contact lines with the

latter. The very special feature of their analysis is that they allowed arbitrary contact angles with the substrate and

did not assume the axial symmetry of perturbations. As the particular case, they determined that σ(k) for a free-

standing wire coincides with σ(k) from Nichols and Mullins’ paper [1]. These authors also found that the substrate

stabilizes the wire, with the wire being most stable for small contact angles. Note that the model [4] is still the base

model, since it does not include any of the multitude of physical eﬀects that typically inﬂuence morphological stability

and evolution at the nanoscale. Some of these eﬀects are the anisotropy of a surface energy (that often results in

faceting [8, 17, 18, 44, 45]), stresses and strains [19], substrate wetting by the ﬁlm [30, 46–49], quantum size eﬀect

[49, 50], and electromigration.

∗Corresponding author. E-mail: mikhail.khenner@wku.edu.

arXiv:2210.15797v2 [cond-mat.mtrl-sci] 3 Apr 2023

The purpose of this paper is to re-visit the analysis in Ref. [4] for the situation of electromigration bias of surface

diﬀusion. In the next section we set up the base electromigration model, which is followed by the analysis in sections

IV A-IV C.

Remark 1. Our physico-mathematical model is ﬁrmly rooted in the classical analysis of morphological evolution of

solid surfaces by surface diﬀusion of adsorbed atoms, i.e. the mobile atoms in the surface layer, as was pioneered by

Mullins [51] and Cahn and co-workers [52]. In the decades since the model was introduced, it has proved its usefulness

and validity countless times, as is evidenced by the publications cited above. This author believes that it makes

little sense to pile up the physical eﬀects until the primary eﬀect of interest (electromigration) is well understood

in the context of morphological stability, evolution, and breakup of nanowires. This paper is the ﬁrst step in that

direction. Notice that in a Mullins-type model an eﬀect is introduced via a term in a chemical potential of adatoms,

i.e. eﬀects are additive [50, 51]. Additional eﬀects may be introduced and analyzed in the future, including some

mentioned above and a more complex ones, such as thermal runaway - a harmful side eﬀect of Joule heating that

may result in a wire meltdown as its radius at the point of rupture tends to zero [53]. Regarding electromigration, a

classical phenomenon in solid state physics, inexplicably, its eﬀect on morphological stability and evolution of wires

was not considered theoretically or computationally. This gap in the knowledge is particularly striking, given that

applied physics and phenomenology of electromigration-driven breakup of nanowires has been a fairly active area of

experiment research in the community of nanoelectronics and device reliability engineers since at least 2004 (the year

of the earliest publication this author was able to ﬁnd; see Ref. [54] for comprehensive review of the progress recently

made).

Remark 2. Studies of charged liquid jets subjected to applied electric ﬁelds proliferate in the theoretical ﬂuid

mechanics community, see e.g. Ref. [55] and the references therein. In Ref. [55] it is shown that the axial electric

ﬁeld will promote the predominance of asymmetric (i.e., non-axisymmetric) instability over the axisymmetric mode,

which causes the bending motion in most experimental observations. Whether similar situation occurs for solid wires

subjected to surface electromigration remains to be seen in a future computational study of morphological evolution

[16–18, 20]. Without electromigration, it was shown that for free-standing wires the axisymmetric instability is

dominant [5] (at least when the surface energy is isotropic).

II. THE MODEL

In this section the base model of wire morphological evolution in the applied electric ﬁeld is set up, starting from the

geometry of a wire on a substrate. Surface energy anisotropy, bulk and contact line stresses, wetting, and other factors

complicating analysis are ignored. Apart from introducing surface electromigration via the term ∇s·[M(φ)Ecos (φ)ey]

[24, 31, 33] in Eq. (1) and the corresponding modiﬁcation in the boundary condition (8), the mathematical statement

of the model is identical to Ref. [4] (see below for the description of all mathematical symbols and physical quantities

seen here). The notations follow closely Refs. [4, 24].

A mere local approximation for the surface electric ﬁeld will be assumed, since it yields the correct result for the

real part, σrof the rate of growth or decay of the perturbation from the initial surface morphology [32]. σrcompletely

characterizes the ﬁlm instability, which is the goal of this paper. Non-local electric ﬁeld via the solution of the

boundary-value problem for the electric potential leads to a non-zero imaginary part of σ[24, 31, 56], i.e. a waves

traveling on the surface, emerge. Analysis of traveling waves is deferred to future work.

It will be seen that the base model admits a semi-analytical solution for neutral stability only, that is, in contrast

to Ref. [4], analytical determination of the instability growth rate (the dispersion relation) is analytically impossible

even for this base model. Valuable insights are gained from the analytical solution of neutral stability. It is expected

that accounting for any additional physical eﬀect will require a fully numerical solution of a (linear) stability problem

(and, of course, of a nonlinear morphology evolution problem).

After a growth process has ended, a section of a cylindrical monocrystalline wire of the radius R0is sitting on a

substrate. A wire surface is making the angle αwith the substrate at both contact lines, with 0 < α < π. A set of

cylindrical coordinates (r, θ, y) is introduced, where the angle θis the angular coordinate measured counter-clockwise

from the x-axis, and yis the coordinate along the wire axis (Fig. 1). The wire and the contact lines are inﬁnite in

the ydirection. A wire of constant radius R0is then described as r=R0for θ0≤θ≤π−θ0, where 0 ≤θ0≤π/2.

The contact lines for this wire are located at θ=θ0and at θ=π−θ0. Also, θ0=π/2−α, if 0 < α ≤π/2, or

θ0=α−π/2, if π/2< α ≤π. This wire is the equilibrium shape. A constant electric ﬁeld E0is applied along the wire

axis (the y-axis), inducing electromigration on the wire surface. Thus the wire surface evolves due to surface diﬀusion

and electromigration. At α=πa full-circular wire is free-standing, i.e. there is no contact with the substrate (in

fact, it is near impossible to grow a full-circular wire on a substrate [11, 12]).

FIG. 1: Cross section of a wire. (a) an equilibrium shape, (b) a perturbed state. The cylindrical coordinate system used to

describe a wire is shown. The yaxis is perpendicular to the xand zaxes. Reproduced from Ref. [4], with the permission of

AIP Publishing.

FIG. 2: A sketch of equilibrium wire shapes at various contact angles.

The goal of the following analysis is to determine the instability of the equilibrium shape, where the contact angle

αis a prescribed constant.

Toward this goal, a wire is perturbed away from the equilibrium shape by an arbitrary perturbation. Let such

perturbed wire be described by r=F(t, θ, y). Also let er,eθ, and eybe the (pairwise orthogonal) unit vectors in

the r,θ, and ydirections, respectively, and let partial derivatives of Fbe denoted by subscripts. With this notation,

the position vector of the surface is written as r= (Fcos θ)ex+ (Fsin θ)ez+yey, the normal velocity is rt·n(where

nis the unit outward normal to the surface), and the mean curvature of the surface is κ=∇·n. We also introduce

the angle φthat quantiﬁes the slope of the surface in yz plane, the lengthscale L, and the timescale τ:Fy= tan φ,

L=R0p(α−sin αcos α)/π,τ=L4kT/(DγΩ2ν). Here, kT is the thermal energy, Dthe surface diﬀusivity of the

adatoms, γthe surface energy, Ω the atomic volume, and νthe surface density of the adatoms. The dimensionless

variables are deﬁned as follows: ˜r=r/L, ˜y=y/L, ˜κ=κL,˜

F=F/L,˜

R=R0/L and ˜

∆=∆/L.

Dropping the tildas, the dimensionless equation describing the evolution of the wire surface reads:

rt·n=∇s·[M(φ){∇sκ+Ecos (φ)ey}],(1)

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EectofElectromigrationonOnsetofMorphologicalInstabilityofaNanowireMikhailKhenner1,21DepartmentofMathematics,WesternKentuckyUniversity,BowlingGreen,KY42101,USA2AppliedPhysicsInstitute,WesternKentuckyUniversity,BowlingGreen,KY42101,USA(Dated:April4,2023)SolidcylindricalnanowiresarevulnerabletoaRayle...

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Eect of Electromigration on Onset of Morphological Instability of a Nanowire Mikhail Khenner1 2

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