Materials and Design 110 (2016) 858–864
`
`Contents lists available at ScienceDirect
`
`Materials and Design
`
`j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s
`
`Predicting multilayer film's residual stress from its monolayers
`C.Q. Guo, Z.L. Pei ⁎, D. Fan, R.D. Liu, J. Gong, C. Sun ⁎
`
`Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China
`
`H I G H L I G H T S
`
`G R A P H I C A L A B S T R A C T
`
`• Residual stress in a multilayer film
`equals the weighted average of
`its
`monolayers' stresses.
`• Relative errors in predicting multi-
`layers' stresses are in the range of 0.2%
`to 10.7% in this paper.
`• Alternating multilayers' stresses gradu-
`ally approach a constant value as its
`monolayer number increases.
`• Si-DLC interfaces can either rise or low-
`er DLC films' residual stresses.
`
`a r t i c l e
`
`i n f o
`
`a b s t r a c t
`
`Article history:
`Received 24 May 2016
`Received in revised form 12 August 2016
`Accepted 15 August 2016
`Available online 17 August 2016
`
`Keywords:
`Residual stress
`Multilayer film
`Diamond-like carbon
`CrN/DLC multilayer
`Cathodic vacuum arc
`
`Multilayer film's residual stress was deduced from Stoney formula. A simple stress formula, which means that
`multilayer residual stress can be given by the weighted average of each monolayer's residual stress, was proposed
`and verified through experiments on gradient diamond-like carbon (DLC) and CrN/DLC multilayers prepared by
`cathodic vacuum arc technology. Typical stress formulas for alternating multilayers were also investigated on
`corresponding DLC multilayers. Multilayer samples, together with monolayers existed in multilayers, were pre-
`pared and studied. Surface profilometry and film stress tester were used to measure films' thicknesses and resid-
`ual stresses, respectively. Cross-sectional morphologies of multilayers were observed by scanning electron
`microscope. Results showed that the proposed stress formula was correct and could provide useful instructions
`on multilayer design. The formula's accuracy of predicting multilayer's residual stress through its monolayers
`was also investigated. In the present paper, relative errors of theoretical values were in the range of 0.2% to
`10.7%, which had a strong relationship with the substrate–film interfaces. In addition, as to alternating multilayer
`film, its residual stress is a constant value as the number of monolayers is even; while this number is odd, mul-
`tilayers' residual stress gets close to the constant value gradually and monotonously.
`© 2016 Elsevier Ltd. All rights reserved.
`
`1. Introduction
`
`Residual stresses in films have been of interest to scientists for a long
`time, especially for multilayers [1–6]. Proper residual stress is good
`to raise films' toughness and adhesion to substrates [7,8], while
`
`⁎ Corresponding authors.
`E-mail addresses: zlpei@imr.ac.cn (Z.L. Pei), csun@imr.ac.cn (C. Sun).
`
`excessively high stress may lead to film failure [9–11]. Therefore, lots
`of researchers tried to take efforts to predict residual stress to design
`multilayers with high performance.
`Numbers of methods for predicting residual stresses in multilayer
`films or structures have been proposed. In the research of Hsueh [12],
`an exact closed-form solution was formulated to predict thermal stress
`in elastic multilayer systems. Strain distribution in the system was
`decomposed into a uniform strain component and a bending strain
`
`http://dx.doi.org/10.1016/j.matdes.2016.08.053
`0264-1275/© 2016 Elsevier Ltd. All rights reserved.
`
`EPL LIMITED EX1008
`U.S. Patent No. 10,889,093
`
`858
`
`

`

`C.Q. Guo et al. / Materials and Design 110 (2016) 858–864
`
`859
`
`Fig. 1. Schematic diagram of a gradient multilayer film.
`
`ti≪ts.
`
`(1) t f ¼ ∑
`i
`(2) Young's modulus and Poisson's ratio of the system composed of
`film and substrate are approximately equal to that of substrate.
`(3) No cracks or delamination occur in the film–substrate system.
`(4) Deformation of substrate is in the range of elastic deformation.
`
`Table 1 shows residual stresses of multilayers as well as substrate's
`radii of curvature with corresponding multilayers. Substrate's radius of
`curvature changes from R0 to RA after layer A is deposited.
`Then, following equations can be obtained, according to Eq. (2).
`
`
`
`1 R
`
`A
`
`σa ¼ K
`ta
`
`1 R
`
`B
`
`σab ¼
`
`K
`
`
`ta þ tbð
`
`Þ
`
`σabc ¼
`
`K
`
`
`ta þ tb þ tcð
`
`Þ
`
`− 1
`R0
`
`
`σabcd ¼
`
`K
`
`
`ta þ tb þ tc þ tdð
`
`Þ
`
`
`
`
`
`
`
`− 1
`R0
`
`− 1
`R0
`
`
`1 R
`
`D
`
`− 1
`R0
`
`
`1 R
`
`C
`
`component. Zhang et al. [13] proposed an analytical model based on
`force and moment balances to predict thermal residual stress distribu-
`tions in multilayered coating systems. A closed-form solution of thermal
`stress was obtained which is independence of the number of coating
`layers. They also used a similar model to predict distribution and mag-
`nitude of thermal residual stress in multilayer coatings with graded
`properties and compositions [14]. Systematical analysis of effects of
`the gradient exponent, elastic modulus of ceramic component, number
`of coating layers and substrate's properties on thermal residual stress
`was conducted on ZrO2Y2O3/NiCrAlY functionally and compositionally
`graded thermal barrier coatings. In another study, Zhang et al. [15] put
`forward a numerical model to predict the thermally induced residual
`stresses in the multilayer coating on a substrate with cylindrical geom-
`etry. This model is based on that the axial forces in the longitudinal di-
`rection and the interfacial pressures in the radial direction, which
`were derived from differential thermal contraction between the adja-
`cent layers, could be determined by the continuity conditions at the in-
`terfaces using layer-by-layer procedure. In addition, finite element
`simulation has also been used to predict residual stresses in thermal
`barrier coatings [16–18]. Though all these methods can help people un-
`derstand residual stress distribution or magnitude within multilayers,
`they either can't be applied to films with high intrinsic stresses or
`can't provide specific value when some parameters involved in the
`model are hard to obtain.
`In the present study, a concise and practical method based on that
`Young's modulus and Poisson ratio of the multilayer-substrate system
`are approximately equal to that of substrate was put forward. A weight-
`ed average formula of multilayer's residual stress derived from Stoney
`formula was presented and then verified by gradient amorphous
`(DLC) as well as composite (CrN/DLC) multilayers. What's more, alter-
`nating multilayer's stress formulas derived from the former weighted
`average formula were also displayed and investigated through corre-
`sponding multilayer DLC films. By analyzing relative errors of theoreti-
`cal values, the formulas' feasibility of predicting multilayer's residual
`stress was discussed in detail.
`
`2. Theory and experimental details
`
`2.1. Theory
`
`When using Stoney formula to calculate residual stress in a film, it is
`considered that thickness of the film is much less than that of substrate
`[4]. The modified Stoney formula can be written as follows:
`
`
`
`;
`
`− 1
`R0
`
`1 R
`
`1 t
`
`f
`
`σ ¼ Et2
`s
`ð
`Þ
`6 1−υ
`
`ð1Þ
`
`where tf is the thickness of film, R0 and R correspond to radii of curvature
`of substrate measured before and after film deposition [19]. As for cer-
`tain substrates, Young's modulus (E), Poisson's ratio (υ) and thickness
`2/[6 (1 −υ)] in Eq. (1) can
`(ts) are all invariable values. So the part Ets
`be replaced by a constant: K. Hence, the modified Stoney formula
`could be rewritten as
`
`
`
`:
`
`− 1
`R0
`
`1 R
`
`σ ¼ K
`t f
`
`ð2Þ
`
`ð3Þ
`
`ð4Þ
`
`ð5Þ
`
`ð6Þ
`
`ð7Þ
`
`ð8Þ
`
`ð9Þ
`
`The first and second assumptions suggest that when layer (i+1) is
`deposited, the original substrate together with layers from 1 to i acts
`as new substrate for layer (i+1). Therefore, residual stresses of mono-
`layers B, C and D can be expressed as
`
`
`
`;
`
`− 1
`RA
`
`
`− 1
`RB
`
`1 R
`
`C
`
`1 R
`
`B
`
`
`
`
`
`:
`
`− 1
`RC
`
`
`
`1 R
`
`D
`
`σb ¼ K
`tb
`
`σc ¼ K
`tc
`
`and
`
`σd ¼ K
`td
`
`Under harsh conditions, films made up by one single layer some-
`times can't meet the needs any more. Fig. 1 illustrates a common design
`of a gradient multilayer film consisting of four kinds of monolayers: A, B,
`C and D. Here, ti and σi represent thickness and residual stress of a cer-
`tain monolayer, respectively. Several assumptions are made for later
`derivation.
`
`Table 1
`Residual stresses of multilayers and corresponding radii of curvature.
`
`Multilayers
`
`Residual stress
`Radius of curvature
`
`AB
`σab
`RB
`
`ABC
`σabc
`RC
`
`ABCD
`σabcd
`RD
`
`

`

`860
`
`C.Q. Guo et al. / Materials and Design 110 (2016) 858–864
`
`According to Eqs. (3) to (9), residual stresses (σab, σabc and σabcd) of
`multilayers can be represented by residual stresses (σa, σb, σc and σd)
`and thicknesses (ta, tb, tc and td) of its monolayers:
`
`;
`
`σab ¼ σ ata þ σ btb
`ta þ tb
`σabc ¼ σ ata þ σ btb þ σ ctc
`ta þ tb þ tc
`σabcd ¼ σ ata þ σ btb þ σ ctc þ σ dtd
`ta þ tb þ tc þ td
`
`;
`
`:
`
`As a result, it can be easily seen from Eqs. (10), (11) and (12),
`Xn
`σ itiXn
`i¼1
`
`σ 1;2; :::n ¼
`
`;
`
`ti
`
`i¼1
`
`ð10Þ
`
`ð11Þ
`
`ð12Þ
`
`ð13Þ
`
`where σ1 , 2 , ... n is residual stress of a multilayer film in which the
`number of monolayers is n (n≥1, and n takes an integer value). That
`is, a multilayer film's residual stress depends upon all its monolayers' re-
`sidual stresses and thicknesses.
`Alternating multilayer film consisting of two kinds of monolayers (A
`and D) with different residual stresses (σa≠σd) illustrated by Fig. 2 is
`often designed to relieve residual stress. Here, m (m ≥ 1, and m takes
`an integer value) represents the number of bilayers in an alternating
`multilayer film. Thus, residual stress of the multilayer film can be repre-
`sented by σ1 , 2 , …2m (the number of monolayers is even) or
`σ1,2, …(2m+1) (the number of monolayers is odd). According to Eq. (13),
`σ 1;2;…2m ¼ mσ ata þ mσ dtd
`¼ σ ata þ σ dtd
`ð14Þ
`mta þ mtd
`ta þ td
`
`;
`
`and
`
`
`
`Þ ¼ m þ 1ð Þσ ata þ mσ dtd
`σ 1;2;… 2mþ1
`m þ 1ð Þta þ mtd
`
`
`ð
`
`:
`
`ð15Þ
`
`From Eqs. (14) and (15), clear conclusions can be drawn. If the num-
`ber of monolayers is even, residual stress of the multilayer is a constant,
`which is only related to its two monolayers' residual stresses and thick-
`nesses and has no connection with the bilayer number m. Otherwise, it
`will vary with m.
`Hence,
`
`σ 1;2;…2m−σ 1;2;… 2mþ1
`ð
`
`Þ ¼
`
`Þtatd
`ð
`σ d−σ a
`Þ m þ 1½ð Þta þ mtd
`
`
`
`
`
`ta þ tdð
`
`Š :
`
`ð16Þ
`
`Fig. 2. Schematic diagram of an alternating multilayer film.
`
`Obviously, the value of (σ1,2, …2m−σ1,2, …(2m+1)) drops monoto-
`nously with the increasing of the bilayer number m, which means that
`extent of variation of residual stress in alternating multilayer film de-
`creases as film thickness rises. As a result, it becomes equal to zero
`when m tends towards infinitude:
`
`
`lim
`m→þ∞
`
`σ 1;2;…2m−σ 1;2;… 2mþ1
`ð
`Þ
`
` ¼ 0:
`
`ð17Þ
`
`Consequently, as to alternating multilayer film, residual stress in
`odd-numbered multilayers changes monotonously with increased
`layer number, and gradually approach the stress value of the even-num-
`bered multilayers.
`
`2.2. Experimental details
`
`Two kinds of multilayer films (gradient multilayer film—sample Ι
`and sample II, alternating multilayer film—sample ΙIΙ) with the structure
`exhibited in Figs. 1 and 2 were deposited on P (100) Si wafers. Samples I
`and III were multilayer DLC films prepared by a filtered cathodic vacu-
`um arc (FCVA) ion-plating apparatus with a 90° bend plasma duct fitted
`between the cathodic arc source and the coating chamber. While in
`sample II consisting of three CrN layers (layers A⁎, B⁎ and C⁎) and one
`DLC layer (layer A), CrN layers were prepared by a direct cathodic vac-
`uum arc (DCVA) source. As to sample ΙIΙ, four bilayers composed of
`layer-A and layer-D stack on Si substrate. The coating equipment had
`been described in detail in previous research [20]. Before film deposi-
`tion, the chamber was evacuated to 4.0 × 10−3 Pa. When depositing
`DLC films high-purity Ar (99.999%) was chosen as working gas to sus-
`tain arc discharge with a working pressure of about 0.1 Pa, while it
`was high-purity N2 (99.999%) with a working pressure of about 0.7 Pa
`for CrN films.
`DLC monolayers A, B, C and D with different residual stresses were
`deposited under different substrate bias voltage (−600 V, −400 V,
`−150 V and −100 V, respectively). Similarly, CrN monolayers A⁎, B⁎
`and C⁎ were prepared also by varying substrate bias voltage (−200 V,
`−400 V and −600 V). Duty ratio and repetition frequency were kept
`invariable (30% and 50 kHz, respectively) throughout deposition proce-
`dure. Different thicknesses of layers were controlled by varying deposi-
`tion time.
`Si wafers with the size of 25 × 5 × 0.4 mm3 were ultrasonically
`cleaned in acetone and ethanol for about 15 min respectively, then sur-
`veyed by film stress tester (FST 150, SPI, China) for radius of curvature
`(R0). Each time before closing the chamber gate, three Si wafers were
`put into the equipment for different purpose. One was for multilayer de-
`position. Another was used to measure residual stress of a monolayer on
`Si substrate. The third one with partly covered by aluminum foil was to
`gain thickness of a monolayer. When the first monolayer was prepared,
`all the samples were taken out of the chamber. Then the two Si wafers
`without aluminum foil were surveyed again by FST 150 for radius of cur-
`vature immediately. Subsequently, another two new clear Si substrates,
`of which one was partly covered by aluminum foil, were put into the
`chamber together with the sample which was used for multilayer depo-
`sition. After that, next monolayer could be prepared. This process was
`repeated until the multilayer film deposition was completed. Finally,
`three multilayer samples (samples Ι, ΙΙ and III), as well as sixteen mono-
`layer samples, were prepared. Throughout the deposition process, no
`plasma etching was performed to avoid affecting film stress and
`thickness.
`Surface profilometry (Alpha-step IQ, KLA Tencor, USA) with a reso-
`lution of 0.0328 nm and a repeatability of 0.1% was used to investigate
`the thicknesses of films. Stylus force, scan length and scan speed were
`29.7 mg, 1000 μm and 50 μm/s, respectively. Step profiles of three points
`were collected for each film. Residual stresses of all films or layers were
`also calculated by FST 150 based on Stoney equation. Biaxial modulus of
`Si wafers is 180.5 GPa [21,22]. The results had a margin of error of plus
`
`

`

`C.Q. Guo et al. / Materials and Design 110 (2016) 858–864
`
`861
`
`or minus 2.0%. Cross sectional morphology and structure of multilayer
`films were observed by field emission scanning electron microscopy
`(SEM, Inspect F50, FEI, USA).
`
`3. Results and discussion
`
`3.1. Gradient multilayer film
`
`3.1.1. Verification and application of the method for amorphous multilayer
`Fig. 3a illustrates the step profiles of gradient DLC multilayer (sam-
`ple I) and monolayers on Si substrates measured by surface
`profilometry. The left part was bare substrate, and the right part was
`covered with film. Thicknesses of films were gained by comparing the
`different heights between the two parts: 570.6 nm (tabcd), 230.3 nm
`(ta), 175.0 nm (tb), 78.7 nm (tc) and 86.6 nm (td). From the cross-sec-
`tional morphology of sample Ι showed in Fig. 3b, it is easy to see that
`four monolayers stacked on Si substrate. No pores or cracks were ob-
`served both inside the monolayers and at the interfaces. It also con-
`firmed the values of thicknesses.
`Gradient multilayer DLC films' residual stresses (σ1,2, ...n) as well as
`that of monolayers (σi) in sample Ι were obtained by FST 150 based
`on Stoney formula through corresponding radius of curvature (R0, RA,
`RB, RC, RD) and thickness (ta, tb, tc, td, tabcd). Values of residual stresses
`are displayed in Table 2. According to Eq. (13), value of residual stress
`in multilayer (σ1,2, ...n) could also be calculated through its monolayers'
`thicknesses (ti, measured by surface profilometry) and residual stresses
`(σi, presented in Table 2). As expected, the two series of results—gaining
`through FST 150 and calculating with Eq. (13)—have the same values
`when their accuracy is three significant figures after decimal point.
`This suggests that the assumption made in Section 2.1 is rational and
`the derivation process of Eq. 13 is correct and logical.
`However, when multilayer film is completed, it is inconvenient and
`meaningless that gaining multilayer's residual stress through its mono-
`layers' thicknesses and stresses instead of calculating directly with Ston-
`ey formula.
`In most cases, thicknesses and residual stresses of monolayers are
`well studied before multilayer film is prepared. Therefore, residual
`stress in multilayer film can be predicted in advance with Eqs. (13),
`(14) or (15), where significance of the formulas lies.
`Application of Eq. (13) in calculating multilayers' residual stresses
`was investigated in detail through residual stresses in monolayers (A,
`⁎). Relative errors (δ) of
`B, C and D) directly deposited on Si wafers (σi
`⁎) were obtained with the following formula:
`theoretical values (σ1,2, ...n
`
`δ=(|σ1,2, ...n⁎−σ1,2, ...n|/σ1,2, ...n)×100%, which were shown in Table 2.
`For a better understanding of the difference between measured re-
`sidual stresses and theoretical values of multilayers, results shown in
`Table 2 are illustrated in Fig. 4. The theoretical values were a little
`lower than measured values but still very close. That means, it is feasible
`to predict multilayer film's residual stress by Eq. (13).
`
`Table 2
`
`
`
`
`⁎).Residual stresses of monolayers (σi, σi⁎) and gradient DLC multilayers (σ1,2, ...n, σ1,2, ...n
`
`Monolayer
`σi (−GPa)
`⁎ (−GPa)
`σi
`Multilayer
`σ1,2, ...n (−GPa)
`⁎ (−GPa)
`σ1,2, ...n
`δ
`
`A
`
`1.555
`1.555
`–
`–
`–
`–
`
`B
`
`1.959
`1.863
`AB
`1.729
`1.688
`2.4%
`
`C
`
`3.455
`2.937
`ABC
`2.010
`1.891
`5.9%
`
`D
`
`4.322
`4.141
`ABCD
`2.361
`2.233
`5.4%
`
`A closer look at Fig. 4 shows that residual stresses of monolayers on
`⁎) are much lower than that of monolayers inSi substrates (σb⁎, σc⁎ and σd
`
`
`
`sample Ι (σb, σc and σd), which leads to previous relative errors showed
`in Table 2. Considering the same deposition parameters for a certain
`monolayer, it is the difference between Si\\C interface and C\\C inter-
`face which leads to the residual stresses of unequal value. Probably,
`the different thermal expansion coefficients and atomic distances be-
`tween DLC and Si substrate as well as mutual diffusions between carbon
`atoms and silicon atoms [23] contribute to the different residual stresses
`of monolayers deposited simultaneously. Moreover, though compres-
`sive stress of layer D is up to 4.322 GPa, that of gradient multilayer
`(ABCD) is 2.361 GPa. That is, introducing intermediate layers is an effec-
`tive method to control film's residual stress.
`
`3.1.2. Verification and application of the method for composite multilayer
`Sample II containing three CrN layers (layer A⁎, B⁎ and C⁎) and one
`DLC layer (layer A) was prepared to check whether Eq. (13) is support-
`ed by crystalline and composite multilayers. Step profiles of the four
`monolayers measured by surface profilometry were presented in Fig.
`5a, in which thicknesses could be easily gained: 174.1 nm (ta⁎),
`107.7 nm (tb⁎), 116.4 nm (tc⁎) and 126.3 nm (ta). Cross-sectional mor-
`phology of sample II was observed by SEM and shown in Fig. 5b. The in-
`terface between DLC and CrN is clear compared with the blurry
`interfaces between CrN monolayers. Thicknesses of layer A and sample
`II presented in Fig. 5b consist with the results shown in Fig. 5a.
`Residual stresses of composite multilayers (σ1,2, ...n) and monolayers
`in sample II (σi) presented in Table 3 were calculated by FST 150 based
`on Stoney formula from corresponding thickness and radius of curva-
`ture. According to Eq. (13), values of σ1,2, ...n could also be obtained by
`residual stresses (σi) and thicknesses (ti) of its monolayers. Multilayers
`A⁎B⁎ and A⁎B⁎C⁎ are crystalline films, while multilayer A⁎B⁎C⁎A (sample
`II) is a kind of composite film. The equality of the two series of results
`with three significant figures after decimal point suggests that Eq.
`(13) is applicable to both crystalline and amorphous multilayers.
`Feasibility of predicting composite multilayers' residual stresses
`⁎)(σ1,2, ...n⁎) from monolayers' thicknesses (ti) and residual stresses (σi
`
`
`deposited directly on Si wafers was also investigated. Relative errors
`⁎ are pre-(δ) of theoretical values (σ1,2, ...n⁎) together with σ1,2, ...n⁎ and σi
`
`
`
`sented in Table 3. The maximum value of relative error is 7.8% which
`
`Fig. 3. (a) Step profiles of gradient multilayer, monolayers A, B, C and D. (b) Cross-sectional morphology of sample Ι.
`
`

`

`862
`
`C.Q. Guo et al. / Materials and Design 110 (2016) 858–864
`
`Table 3
`
`
`
`
`⁎).Residual stresses of monolayers (σi, σi⁎) and composite multilayers (σ1,2, ...n, σ1,2, ...n
`
`Monolayer
`σi (−GPa)
`⁎ (−GPa)
`σi
`Multilayer
`σ1,2, ...n (−GPa)
`⁎ (−GPa)
`σ1,2, ...n
`δ
`
`A*
`
`2.735
`2.735
`–
`–
`–
`–
`
`B*
`
`3.03
`3.044
`A⁎B⁎
`2.848
`2.853
`0.2%
`
`C*
`
`2.421
`2.809
`A⁎B⁎C⁎
`2.723
`2.84
`4.3%
`
`A
`
`1.001
`1.379
`A⁎B⁎C⁎A
`2.308
`2.488
`7.8%
`
`sample ΙIΙ were raised compared with that of sample I, correctness of Eq.
`(13) is still well verified.
`
`3.2.2. Application of the method
`Feasibility of application of Eqs. (14) and (15) in calculating alternat-
`ing multilayers' residual stress was also studied. Residual stresses of
`⁎,monolayers on Si substrates (σi⁎) and alternating multilayers (σ1,2, ...n
`
`
`which was calculated with Eqs. (14) and (15)) are illustrated in Table
`4. To reduce the influence of experimental error, ta and σa in Eqs. (14)
`and (15) refer to average thickness (86.8 nm) and residual stress
`(−1.369 GPa) of monolayers (A) on Si substrates, while td (86.5 nm)
`and σd (−4.437 GPa) correspond to that of monolayers (D). Relative er-
`⁎ were also calculated and then presented in Table 4.
`rors (δ) of σ1,2, ...n
`In order to get a deeper insight into the evolution of alternating mul-
`tilayers' residual stress versus thickness, values in Table 4 are showed in
`Fig. 7. The difference between theoretical values and measured values is
`very small. In addition, compressive stresses of even-numbered multi-
`layers are very close to the theoretical value (2.9 GPa). While compres-
`sive stresses of multilayers ADA, 2(AD)A and 3(AD)A rise with the
`increased layer number, approaching this theoretical value (2.9 GPa)
`gradually. This result is well consistent with the derivation in Section
`2.1. Lots of researchers think that stress relaxation by multilayer struc-
`turing is due to the view plastic deformation is easy to occur for the
`monolayer with lower stress [24–26]. Eqs.(14) and (15) provide a the-
`oretical support for this explanation. Another interesting phenomenon
`is that as for monolayers A2, A3 and A4, compressive stresses of mono-
`layers in sample ΙIΙ are always lower than that of monolayers on Si sub-
`strates. While for monolayers D1, D2, D3 and D4, the opposite is the
`case. That is, Si-DLC interface can either rise or lower residual stress in
`DLC film.
`Before this paper, lots of researchers have investigated residual
`stresses in alternating multilayer films. Table 5 presents the measured
`values (σ1,2, ...2m) reported in literatures and the theoretical values cal-
`⁎). It's obvious to see, though bilayer
`culated with Eq. (14) (σ1 ,2 , ... 2m
`number (from 2 to 30), thickness ratio (from 1:1 to 1:5) and mono-
`layers' stress (from −0.8 to −13.7 GPa) varied a lot, theoretical values
`
`Fig. 4. Evolution curve of residual stress versus thickness for gradient multilayer film.
`
`originates from the different residual stresses in monolayers whether
`they were deposited directly on Si wafers or existed in multilayer
`films. These small relative errors showed that predicting residual stress
`in composite multilayers according to Eq. (13) could be realized.
`
`3.2. Alternating multilayer film
`
`3.2.1. Verification of the method
`To differentiate each layer in sample ΙIΙ, the monolayers were named
`as A1, D1, A2, D2, A3, D3, A4 and D4 according to deposition order. Mul-
`tilayers of ADAD, ADADA, ADADAD, ADADADA and ADADADAD can be
`abbreviated as 2(AD), 2(AD)A, 3(AD), 3(AD)A and 4(AD). Step profiles
`of alternating multilayer 4(AD) and monolayers (layer A2 and D2) are
`displayed in Fig. 6a. Thickness of 4(AD) is 693.2 nm. Fig. 6b presents
`cross-sectional morphology of sample ΙIΙ, in which the boundary from
`layer A to layer D is much clearer than that from layer D to A.
`Thicknesses (ti) and residual stresses (σi) of monolayers in sample ΙIΙ
`as well as alternating multilayers' residual stresses (σ1 , 2 , ... n) are
`displayed in Table 4, in which residual stresses were obtained by FST
`150 based on Stoney formula. Similar to the results of gradient multilay-
`er, alternating multilayers' residual stresses calculated with Eq. (13)
`through monolayers' thicknesses (ti) and residual stresses (σi) are the
`same with values (σ1,2, ...n) in Table 4 when they have three significant
`figures after decimal point. Though both thickness and layer number of
`
`Fig. 5. (a) Step profiles of monolayers A⁎, B⁎, C⁎ and A. (b) Cross-sectional morphology of sample ΙI.
`
`

`

`C.Q. Guo et al. / Materials and Design 110 (2016) 858–864
`
`863
`
`Fig. 6. (a) Step profiles of multilayer 4(AD), monolayers A2 and D2. (b) Cross-sectional morphology of sample III.
`
`ð18Þ
`
`4. Conclusions
`
`A new method for assessment of residual stress in multilayer film
`derived from Stoney formula has been put forward and verified by gra-
`dient as well as alternating multilayer films, which provides a feasibility
`of predicting multilayer film's residual stress without doing real exper-
`iments and helps operators optimize multilayer's design.
`The proposed multilayer stress formulas are on the base that Young's
`modulus and Poisson ratio of the system consisting of substrate and film
`are approximately equal to that of substrate. A multilayer film's residual
`stress can be given by the weighted average of residual stresses in its
`monolayers. As to alternating multilayer films, if the number of mono-
`layers is even, its residual stress is a constant value; while the number
`of monolayers is odd, its residual stress rises or declines monotonously
`with the increasing layer number, getting close to the former constant
`value gradually. That means, alternating multilayer film's residual stress
`does not grow infinitely with film deposition, but tends to a constant.
`However, it has restrictions when predicting a multilayer film's residual
`stress through monolayers on Si wafers or other kinds of substrates ac-
`cording to multilayer stress formulas. Relative errors of theoretical
`values mainly come from the difference between monolayers deposited
`directly on substrates and corresponding monolayers existing in multi-
`layer film. In other words, the smaller this difference is, the more accu-
`rate the weighted average formula of multilayer stress is.
`
`were still very close to the values measured in these studies. All relative
`errors are lower than 10%.
`In addition, refs. [24,28,29] also plotted evolution curves of residual
`stress versus film thickness for alternating multilayer films, which
`displayed similar tendency to that showed in Fig. 7. With the increase
`of monolayer number, residual stress in alternating multilayer film got
`close to a constant gradually. This provides more evidences that the for-
`mulas proposed in Section 2.1 not only present new methods to calcu-
`late residual stress in multilayer film (which equals the weighted
`average of its monolayers' residual stresses) but also help researchers
`understand multilayer's residual stress more deeply (how it changes
`as layer number increases).
`For a multilayer film, Young's modulus
`Xn
`EitiXn
`
`E f ¼
`
`i
`
`;
`
`ti
`
`i
`
`where Ei is a monolayer's Young's modulus [30]. According to the
`critical thickness formulas [31]
`t max ¼ 2E f γ f
`σ 2
`
`ð
`Þ
`tensile‐stressed film
`
`ð19Þ
`
`and
`
`
`t max ¼ E f γ f þ γs
`
`σ 2
`
`
`
`ð
`
`Þ;
`compressive‐stressed film
`
`ð20Þ
`
`where γf is surface free energy of the film, γs corresponds to surface
`free energy of the substrate, the rough maximum thickness (tmax) of
`the film without cracking or spalling can be calculated. This presents
`useful information to experimenters on how to deposit multilayer
`films with proper thicknesses and improve their research plan.
`
`Table 4
`⁎) of monolayers as well as alternating multi-
`Thicknesses (ti) and residual stresses (σi, σi
`⁎).
`layers' residual stresses (σ1,2, ...n, σ1,2, ...n
`
`Monolayer
`
`A1
`
`D1
`
`A2
`
`D2
`
`A3
`
`D3
`
`A4
`
`D4
`
`ti (nm)
`σi (−GPa)
`⁎ (−GPa)
`σi
`Multilayer
`σ1,2, ...n
`(−GPa)
`⁎
`σ1,2, ...n
`(−GPa)
`
`δ
`
`80.2
`87.7
`93.0
`81.3
`89.4
`88.3
`83.5
`89.8
`4.947
`0.651
`4.399
`0.927
`1.019 4.775 0.844 4.677
`4.562
`1.377
`4.147
`1.683
`1.019 4.473 1.395 4.567
`–
`AD
`ADA
`2(AD) 2(AD)A 3(AD) 3(AD)A 4(AD)
`–
`2.828 2.159 2.800
`2.448
`2.793
`2.487
`2.772
`
`–
`
`–
`
`2.900 2.389 2.900
`
`2.594
`
`2.900
`
`2.681
`
`2.900
`
`2.6%
`
`10.7% 3.6%
`
`6.0%
`
`3.8%
`
`7.8%
`
`4.6%
`
`Fig. 7. Evolution curve of residual stress versus thickness for alternating multilayer film.
`Squares linked by a solid line represent measured residual stresses of multilayers; circles
`linked by a dash-dotted line represent theoretical values calculated from Eqs. (14) and
`(15). Horizontal solid lines represent residual stresses of monolayers in sample III;
`horizontal dash-dotted lines represent residual stresses of monolayers on Si substrates.
`
`

`

`864
`
`C.Q. Guo et al. / Materials and Design 110 (2016) 858–864
`
`Bilayer number (m)
`
`Alternating multilayer
`
`Thickness ratio (ta:td)
`
`Refs.
`
`[27]
`
`[25]
`[24]
`[26]
`[28]
`[29]
`
`30
`15
`2
`3
`3
`3
`8
`
`Table 5
`⁎).
`Measured values of residual stresses in some previous studies (σ1,2, ...2m) and theoretical values calculated according to Eq. (14) (σ1,2, ...2m
`Residual stress ratio (σa:σd)
`σ1,2, ...2m (−GPa)
`1.6:8.2
`4.5
`5.5:8.2
`7.0
`0.8:4.5
`3.6
`1.0:7.7
`4.5
`4.0:13.7
`8.5
`1.9:6.2
`4.8
`2.0:5.6
`5.2
`
`(soft/hard) DLC
`
`(soft/hard) DLC
`(soft/hard) DLC
`(TiC/DLC)
`(soft/hard) DLC
`(soft/hard) DLC
`
`1:1
`
`1:5
`1:1
`1:1
`1:2
`5:23
`
`⁎ (−GPa)
`σ1,2, ...2m
`4.9
`6.9
`3.9
`4.4
`8.9
`4.8
`5.0
`
`Relative error (δ)
`
`8.9%
`1.4%
`8.8%
`2.2%
`4.7%
`0
`3.8%
`
`Acknowledgments
`
`This work was supported by the National Key Basic Research Pro-
`gram of China (973 Program, No. 2012CB625100) and the Natural Sci-
`ence Foundation of Liaoning Province of China (No. 2013020093).
`
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