Hamed Haftbaradaran

Sumit K. Soni

Brian W. Sheldon

School of Engineering,

Brown University,

Providence, RI 02912

Xingcheng XiaoGeneral Motors Global Research &

Development Center,

30500 Mound Road,

Warren, MI 48090

Huajian Gao1

School of Engineering,

Brown University,

Providence, RI, 02912

e-mail: Huajian_Gao@Brown.edu

Modified Stoney Equation forPatterned Thin Film Electrodeson Substrates in the Presenceof Interfacial SlidingMechanical stresses and failure are believed to be a major cause for the limited cycle lifeof lithium-ion batteries employing high capacity Si electrodes. Recent experiments haveshown that patterned Si thin film electrodes on substrate exhibit improved cycling stabil-ity and substantial sliding at the film/substrate interface. To facilitate experimental stud-ies of stress evolution in such systems, we have developed a modified Stoney equationwhich accounts for the effect of interfacial sliding on the relationship between curvatureand stress in patterned thin films on substrate. [DOI: 10.1115/1.4005900]

Keywords: patterned thin films, rechargeable batteries, silicon anodes, Stoney equation

1 Introduction

Silicon (Si) is a highly promising material for application ashigh capacity negative electrodes in lithium-ion (Li-ion) batteries[1]. However, lithiated Si undergoes huge volume expansion (upto 400%), leading to cracking and degradation when deformationis constrained, thereby reducing the cycle life of batteries using Sielectrodes [2]. To circumvent this issue, much experimental efforthas been devoted to strategies to improve the cycling stability ofSi electrodes, such as manufacturing them into nanowires [3] andcomposites (see Ref. [1] for an extensive review, and Ref. [4] fora recent study).

Recently, patterned Si arrays on copper (Cu) have been used as amodel system to investigate the stability of Si thin film electrodes[5]. As a result of the lithiation/delithiation cycles, continuous Sifilm with thickness of 100 nm exhibited typical “mud-crack” pat-terns with fracture spacing on the order of 5–10 lm. In comparison,patterned Si islands with the same thickness of 100 nm but havinglateral dimensions below the observed crack spacing in a continu-ous film adhered to the Cu substrate after 30 cycles, while thosehaving larger lateral dimensions delaminated away. This phenom-enon has been attributed to the initiation of plastic deformation inSi at a critical island size during lithiation/delithiation [5]. It istherefore of great interest to design Si thin film arrays such that themaximum stress in the electrodes during cycling stays below theflow stress of lithiated Si which has been reported to be about 1 to1.7 GPa [6]. Since the shear yield stress of copper (40 MPa) andthe interfacial sliding strength are two to three orders of magnitudelower than the flow stress of lithiated Si (e.g., the friction strengthof Au/mica interface was measured to be �40 MPa [7]), it can beexpected that the stress distribution in patterned Si islands could bestrongly non-uniform, with the maximum stress controlled by theisland size.

The patterned Si thin film islands on substrate provide a con-venient model system for in situ studies of stress evolution in theelectrodes by monitoring substrate curvature during cycling [8].In this technique, an incident array of parallel laser beams is

directed onto a polished surface of substrate and the reflected sig-nals are captured by a charged coupled device (CCD) camera(e.g., [9]), allowing direct measurements of the substrate curva-ture. Subsequently, Stoney formula is used to relate the measuredcurvature to the stress in the film. However, for Si thin electrodeson substrate, there exists substantial evidence for relative slidingat the film/substrate interface within shear lag zones around theisland edges. For example, Fig. 1 shows an array of patterned Siislands with width of 17 lm and thickness of 50 nm undergoingin-plane (as well as out-of-plane) expansion/contraction on a Ticurrent collector during lithiation/delithiation. It is observed thatthese Si islands undergo later expansions as large as 30%, whichcan only occur in the presence of substantial interfacial sliding.Therefore, a modified Stoney formula that accounts for the effectof interfacial sliding will be needed to relate curvature and stressin such systems. Previously, the Stoney formula has been modi-fied in various ways to account for effects such as non-uniformmismatch strain and material properties [10] as well as geometri-cally nonlinear deformation [11]. In these studies, it has beenassumed that the normal stress remains nearly uniform in the film,and the shear stress along the film/substrate interface near theedge of the film is localized to a very narrow region on the orderof the film thickness. In the case of patterned Si islands on metal-lic current collectors (e.g., Cu or Ti), these assumptions need to berelaxed in view of the interfacial sliding shown in Fig. 1. In suchsystems, the stress in the film can vary over a characteristic lengthscale much larger than the film thickness due to sliding at thefilm/substrate interface [5]. The assumption of constant stress inthe film is no longer valid and a modified Stoney formula thatproperly takes into account stress variation in the film is neededbefore one can relate the experimentally measured curvature datato the maximum stress in the patch, which can be nearly twoorders of magnitude larger than the interfacial sliding strength [5].

In this paper, we develop a modified Stoney equation specificallyaimed for applications to stress measurements in patterned thin filmislands on substrates that permit interfacial sliding. We start with atwo-dimensional analysis of this problem in Sec. 2 and extend it tothree-dimensional cases of circular and square islands on substratein Sec. 3. In Sec. 4, the modified Stoney formula is used to interpreta set of in situ curvature measurements in patterned Si islands. Asummary of the results is presented in Sec. 5.

1Corresponding author.Manuscript received August 7, 2011; final manuscript received December 19,

2011; accepted manuscript posted February 13, 2012; published online April 4,2012. Editor: Robert M. McMeeking.

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2 Two-Dimenisonal Modifed Stoney Equation

We start by analyzing the two dimensional plane-strain problemof an array of thin film islands, each with length 2l, thickness hf

and edge-to-edge gap 2s on a substrate of thickness hs, as shownin Fig. 2. The Young’s moduli of the film and substrate aredenoted as Ef and Es, respectively. Taking advantage of the sym-metries in the problem, we focus on a half period of the structure

with length lþ s, located between section “A” passing throughthe midpoint of one patch and ending at section “B” bisecting thegap between the patch and its neighboring patch (Fig. 2(a)).Figure 2(b) shows the free-body diagram of the portion of sub-strate between the two sections. To obtain the curvature inducedin the substrate due to the stress in the film, we consider a virtualmoment M at section “B” and invoke Betti’s reciprocity theoremas follows. The total work done (per unit out-of-plane depth) bythe interfacial stresses sintðxÞ through displacement induced by Mis equal to the work done by the moment M through rotation hsinduced by the islands:

M hs ¼ðl

0

sintðxÞ dMðxÞ dx (1)

where dMðxÞ denotes the horizontal displacement at the interfaceinduced by M. From the classical beam theory, the axial strain

induced by M at the surface of the substrate is ex ¼ M hs

2EsIs, where

Is ¼ 112

h3s is the moment of inertia of the substrate (per unit out-

of-plane depth). Hence

Fig. 1 Optical micrographs of Si islands on a Ti substrate,indicating substantial in-plane expansion/contraction duringlithiation and delithiation. (a) The original size of the islands isaround 17 lm; (b) under lithiation, the Si islands expand byroughly 30% in the plane of the substrate, indicating substantialinterfacial sliding, and (c) after delithiation, the Si islands areseen returning towards the original size. The scale bar is 50 lm.

Fig. 2 Modified Stoney equation for two-dimensional pat-terned films on substrate. (a) The geometry of the problem, and(b) the reciprocity theorem for determining the curvature of thesystem; (c) comparison of FEM calculations and theoreticalpredictions of curvature for two-dimensional patterned films onsubstrate.

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dMðxÞ ¼ exx ¼ 6M x

Esh2s

(2)

On the other hand, the rotation hs is related to the average curva-ture of the substrate j via

hs ¼ �j ðlþ sÞ (3)

where the sign convention is such that tensile stress in the filmcauses positive curvature.

Substituting Eqs. (2) and (3) into Eq. (1), and introducing thedimensionless parameters ~x ¼ x

l, ~sintð~xÞ ¼ sintðxÞ=s0, where s0 isthe interfacial sliding strength, the curvature j can be expressed as:

j ¼ � 6s0l

Esh2s

l

lþ s

ð1

0

~sintð~xÞ ~x d~x (4)

To compute the integral on the right hand side of Eq. (4), oneneeds to know the distribution of shear stress along the interface.Assuming that the film is thin enough to be treated as a membraneand that it would slide freely on the substrate as soon as the shearstress along the film/substrate interface attains s0, it can be shownthat sintðxÞ should equal s0 within a distance lp from the edge andthen drops to zero beyond this region, i.e.,

sintðxÞ ¼ �s0 signðrf Þ for l > x > l� lp0 for l� lp > x > 0

�(5)

A simple equilibrium consideration would show that the size ofthe sliding zone is lp ¼ rf

�� ��hf

�s0. Using Eq. (5) along with this

expression for lp, Eq. (4) then yields the following modifiedStoney equation for curvature:

j ¼ 6 s0 l

Esh2s

l

lþ s~rf 1� ~rf

2

� �signðrf Þ (6)

where ~rf ¼rfj jhf

s0l . In the limit of ~rf � 1, Eq. (6) reduces to the

conventional Stoney formula j ¼ 6 rf hf

Esh2s

llþs.

In order to check the validity of the two-dimensional modifiedStoney formula in Eq. (6), we have performed finite element(FEM) calculations via ABAQUS to determine the curvatureinduced by a uniform thermal strain in an array of thin filmpatches on substrate. Three different patch spacings s¼ 1, 3, and6 lm were considered. The Young’s modulus and Poisson ratio ofthe patch are taken as 40 GPa and 0.22 (which are close to the val-ues reported for lithitated Si [12]), and those of the substrate as130 GPa and 0.34 (as given for copper by [9]). The patch lengthand thickness are taken to be 6 lm and 100 nm, respectively, andthe substrate 50 lm thick. A rigid-plastic cohesive model isadopted to characterize the interfacial slipping under a slidingstrength of 40 MPa. The results are shown in Fig. 2(c) along withthe theoretical prediction from the conventional Stoney formulaas well as that from the modified Stoney formula in Eq. (6). The

dimensionless parameter ~j ¼ j Esh2s

6 s0llþs

l has been introduced. The

predicted values of curvature from Eq. (6), shown as the solid linein Fig. 2(c), are in good agreement with the calculated resultsfrom FEM calculations. In contrast, the conventional Stoney for-mula exhibits up to 50% underestimate of the maximum stress inthe film.

3 Three-Dimensional Modified Stoney Equations

In this section, we will derive a modified Stoney formula forcircular islands on substrate and then extend it to the more diffi-cult case of square islands with the aid of FEM simulations.

3.1 Circular Islands. The approach taken in the precedingsection may not be generally applicable to three-dimensional

islands. For an array of circular islands, we consider a quarter ofthe unit cell of the periodic structure, as shown in Fig. 3(a), andassume the stress within the island remains approximately axi-symmetric. Following a procedure based on Betti’s reciprocitytheorem similar to the two-dimensional case, it can be shown thatthe curvature induced in the substrate can be written in the follow-ing form:

j ¼ � 6s0Rð1� �sÞEsh2

s

p R2

4 Rþ sð Þ2ð1

0

~sintð~rÞ ~r2 d~r (7)

where R is the radius of the island, s is the half-spacing betweenislands, ~r ¼ r

R and �s is the Poisson ratio of the substrate. Com-pared to the two-dimensional case, the factor 1� �s is a resultof the three-dimensional geometry. To calculate the integral inEq. (7), we again assume the following shear stress distribution atthe interface

sintðrÞ ¼ �s0 signðrf Þ for R > r > rp

0 for rp > r > 0

�(8)

in the limit of a very thin film. It follows that Eq. (7) can be inte-grated to the following modified Stoney equation,

Fig. 3 Modified Stoney equation for three-dimensional pat-terned films on substrate, and comparison with FEM calcula-tions. (a) The geometry of the problem, and (b) the reciprocitytheorem for determining the curvature of the system.

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j ¼ 6s0Rð1� �sÞEsh2

s

p R2

4 Rþ sð Þ21

3�

~r3p

3

!signðrf Þ (9)

where, unlike the two-dimensional case, ~rp ¼ rp

R cannot bedetermined from equilibrium condition alone, but needs tobe determined from both equilibrium and continuity conditions inthe island. To determine ~rp, the patch is considered as a plane-stress circular disk subject to a body force corresponding to theinterfacial shear stress in the original problem. The axisymmetricdeformation in the island is described by the following differentialequation

d

dr

1

r

d

drr uð Þ

� �þ

bð1� �2f Þ

Ef¼ 0 (10)

where u is the radial displacement and b is the body force actingin the radial direction,

b ¼ 1

hf

s0 signðrf Þ for R > r > rp

0 for rp > r > 0

�(11)

Now assume that the patch is subject to a uniform isotropic mis-match strain e�, as a result of Li insertion or other causes. The ra-dial component of stress is related to the displacement via

rrr ¼Ef

1� �2f

du

drþ �f

u

r� ð1þ �f Þe�

� (12)

Within the region 0 � r � rp, the differential Eq. (10) and theboundary conditions u ¼ 0 at both r ¼ 0 and r ¼ rp indicates thatthe stress in this region of the island is uniform,

rf ¼ �Ef

1� �fe� (13)

which is also the maximum stress in the island as a whole. Withinthe sliding region rp � r � R, Eq. (10) can be solved subject toboundary conditions u ¼ 0 at r ¼ rp and rrr ¼ 0 at r ¼ R , andfinally rp can be obtained in terms of e� from the continuity of rrr

at r ¼ rp. Using Eq. (13) to replace e� with the maximum stressrf in the island, we find that the result can be expressed in termsof the following dimensionless form:

~rp ¼ ð1þ �f Þð1� �f Þ�2=3A�1=3 � ð1� �f Þ�1=3A1=3 (14)

where

A ¼ ð3~rf � 2� �f Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9~r2

f � 6ð2þ �f Þ~rf þ5þ 3�f

1� �f

s(15)

and ~rf ¼rfj jhf

s0R . Thus, for a given maximum stress rf , ~rp is given by

Eqs. (14), (15), and the curvature can be obtained from Eq. (9). Itshould be pointed out that Eq. (9) in conjunction with Eq. (15)

reduces to the conventional Stoney formula j ¼ 6rf hf ð1��sÞEsh2

s

p R2

4 Rþsð Þ2 in

the limit of ~rf � 1. Figure 4(a) shows the results of FEM calcula-tions for three different cases with s ¼ 1, 3, 6 lm and R ¼3 lm,along with the theoretical predictions from the above analysisand from the conventional Stoney formula, for the normalized cur-

vature ~j ¼ j Esh2s

6 ð1� �sÞs0R4p

R þ sR

� �2. Numerical values for other parame-

ters are the same as those adopted for the two dimensional case. Itis seen that, while the theoretical predictions from Eqs. (9), (14),and (15) are in fair agreement with the FEM simulations, the con-ventional Stoney formula could lead to significant underestimate ofthe maximum stress in the film.

3.2 Square Islands. For the case of square islands, the shearstresses along the interface seem too complicated for an analyticalstudy. For a simplified treatment, the square patches can be approxi-mated as an equivalent array of circular patches which induce thesame curvature in the substrate at the same level of maximum stress.To this end, we rewrite Eq. (9) in the following form

j ¼ 6s0Reqð1� �sÞEsh2

s

l

lþ s

� �21

3�

~r3p

3

!sign ðrf Þ (16)

Here, we assume that ~rp is still obtained through Eqs. (14) and

(15), but with dimensionless stress ~rf ¼rfj jhf

s0l . Any change in the

form of curvature dependence on the maximum stress is thenlumped into an equivalent radius of the patch, which from a

dimensional analysis should be in the form Req ¼ l ~Reqð~rf Þ. ForEq. (16) to be reduced to the conventional Stoney formula,~Reqð~rf Þ must tend to unity as ~rf approaches zero. With this as-

ymptotic behavior, we propose that ~Reqð~rf Þ ¼ 1þ a ~rbf where a

and b are constants to be obtained from fitting Eq. (16) to a set of

FEM calculations. Our calculations indicate that a ¼ 14

and b ¼ 12

provide an acceptable fit, as shown in Fig. 4(b).

4 Interpretation of Experimental Data

The modified Stoney formula developed in the present paperhas been used to interpret a set of in situ stress measurementsshown in Fig. 5. Note that the nominal stresses shown in this fig-ure are obtained using the conventional volume-averaged Stoneyequation and the initial film thickness. The specimens consisted ofSi islands with initial dimensions of l¼ 20 lm, s¼ 9 lm, and

Fig. 4 Comparison of FEM calculations and theoretical predic-tions of curvature for (a) circular patches where the solid line isthe theoretical prediction; (b) square patches where the solidline is based on a fitting scheme

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hf ¼ 50 nm, prepared by through mask sputtering on a 400 nmthick Ti current collector (which was deposited on a 230 lm thickquartz glass substrate). The fabrication methods and stress meas-urements have been described in more details elsewhere [5,13].The specimens were cycled at a rate of C/20, to progressivelylower voltages in each subsequent cycle. The results in Fig. 5(a)were obtained by running to 0.5 V during the first cycle and 0.4 Vduring the second cycle (only the second cycle is shown here),and those in Fig. 5(b) were obtained after cycling further to 0.3 Vand 0.2 V during the third and fourth cycles, respectively (onlythe fourth cycle is shown in Fig. 5(b)). Based on the total meas-ured current, the amount of Li inserted/removed in these cycleswas 505/270 mAh/g, 350/331 mAh/g, 470/464 mAh/g and800/760 mAh/g, respectively. The differences between the inser-tion and removal capacities are probably associated with the for-mation of solid-electrolyte interphase (SEI), which is morepronounced in the first cycle. Decreasing the voltage at the end ofthe Li insertion step allows more Li insertion in each subsequent

cycle. A more detailed description of the method used to compareour analysis with the experimental results is given elsewhere [8].Note that s0 is the only quantity in Eq. (16) that cannot beobtained directly from the literature or the experimental configu-ration. Adjusting this value to fit the experiments gives excellentagreement with the data in Fig. 5(a), with s0 ¼ 12:2 MPa. Thisvalue is somewhat smaller but comparable to the approximatevalue which was previously inferred indirectly from fracture pat-terns in lithiated Si on Cu [5]. In the present study, the analysisindicates that lp ¼ 3:84 lm, and rf ¼ �0:8 GPa (middle of island)at the capacity of 300 mAh/g which is significantly larger than thestress obtained based on the volume-averaged Stoney equation.During the fourth cycle shown in Fig. 5(b), the lower voltage cutoff leads to more lithiation. The fit uses the same value of theinterfacial shear stress obtained above to predict the maximumstress in the island at the onset of flow. Based on the experimentaldata, this occurs at a capacity of 402 mAh/g (see Fig. 5(b)). Thefit shown here is based on a purely elastic response of the island,and agrees with the experiments for only the first portion of thestress data. At the capacity of 402 mAh/g, the calculation plottedin Fig. 5(b) gives lp ¼ 5:18 lm and rf ¼ �1:05 GPa (middle ofisland).

While the stress data, shown in Fig. 5(b) based on the volume-averaged Stoney formula, exhibit flow at �0.52 GPa, the modifiedequations presented in this work suggest that the stress in the cen-ter of island, i.e., �1.05 GPa, is large enough to initiate plasticityin the Si. This stress value is in fact in good agreement with theflow stress in continuous Si films [6,13]. As lithiation proceedspast this point (i.e. the onset of flow), the measured nominalstresses are significantly smaller than the predicted values, whichis consistent with the idea that the stress is now limited by flow inSi. Further confirmation of plastic flow is provided by theobserved hysteresis in the stress data (i.e., between the Li insertionand Li removal portions of the cycle). Note that this type of hys-teresis is not observed in Fig. 5(a), where we expect to see elasticdeformation only, and the higher voltage limit leads to less lithia-tion and the predicted rf value does not reach the flow stress.

5 Summary

We have developed a class of modified Stoney equations whichrelates stress and curvature induced by an array of thin filmislands on substrate in the presence of sliding at the island/sub-strate interface. The interfacial sliding causes large scale in-planestress variations in the islands, in which case the conventionalStoney formula is no longer applicable. The validity of our resultswas verified by comparison with FEM calculations. This work isexpected to be particularly applicable to experimental measure-ments of stress evolutions in patterned Si thin film electrodes inLi-ion batteries.

Acknowledgment

This work has been supported by the GM/Brown CRL on Com-putational Materials Science and National Science Foundationthrough the MRSEC Program (Award No. DMR-0520651). SKSand BWS also acknowledge support from NSF (Award No.CMMI-1000822) and KIST.

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Fig. 5 Stress evolution obtained according to the volume-averaged Stoney equation versus capacity. The modifiedStoney equation in Eq. (16) has been used in the solid curvesand to estimate the maximum stress in the center of island. (a)For the second cycle; the predicted stress at the center of theisland is 20.8 GPa at the capacity of 300 mAh/g and (b) for thefourth cycle; the predicted stress at the center of island is21.05 GPa at the capacity of 402 mAh/g. The deviation of the fit-ted curve form the observed values in part (b) is attributed toplastic flow in the island.

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