simultaneous determination of the residual stress elastic modulus density and thickness of ultrathin film utilizing vibrating doubly clamped micro nanobeams

Simultaneous determination of the residual stress, elastic modulus, density and thickness of ultrathin film utilizing vibrating doubly clamped micro-/nanobeams Ivo Stachiv,1,2, aChih-Yun

Simultaneous determination of the residual stress, elastic modulus, density and thickness of ultrathin film utilizing vibrating doubly clamped micro-/nanobeams

Ivo Stachiv, Chih-Yun Kuo, Te-Hua Fang, and Vincent Mortet

Citation: AIP Advances 6, 045005 (2016); doi: 10.1063/1.4947031

View online: http://dx.doi.org/10.1063/1.4947031

View Table of Contents: http://aip.scitation.org/toc/adv/6/4

Published by the American Institute of Physics

Simultaneous determination of the residual stress, elastic modulus, density and thickness of ultrathin film utilizing vibrating doubly clamped micro-/nanobeams

Ivo Stachiv,1,2, aChih-Yun Kuo,3Te-Hua Fang,1and Vincent Mortet2

1Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan

2Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic

3Tzu-Chi University, Hualian City, Hualian, Taiwan

(Received 11 February 2016; accepted 4 April 2016; published online 12 April 2016)

Measurement of ultrathin film thickness and its basic properties can be highly challenging and time consuming due to necessity of using several very sophisticated devices Here, we report an easy accessible resonant based method capable to simultaneously determinate the residual stress, elastic modulus, density and thick-ness of ultrathin film coated on doubly clamped micro-/nanobeam We show that

a general dependency of the resonant frequencies on the axial load is also valid for in-plane vibrations, and the one depends only on the considered vibrational mode As a result, we found that the film elastic modulus, density and thickness can be evaluated from two measured in-plane and out-plane fundamental resonant frequencies of micro-/nanobeam with and without film under different prestress forces Whereas, the residual stress can be determined from two out-plane (in-plane) measured consecutive resonant frequencies of beam with film under different prestress forces without necessity of knowing film and substrate properties and dimensions Moreover, we also reveal that the common uncertainties in force (and thickness) determination have a negligible (and minor) impact on the determined film properties The application potential of the present method is illustrated on the beam made of silicon and SiO2 with deposited 20 nm thick AlN and 40 nm thick Au thin films, respectively C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).[http://dx.doi.org/10.1063/1.4947031]

I INTRODUCTION

Functional polymer and solid ultrathin films are of emerging interest for variety applications including solar cell panes, biosensors, sensors for the environmental monitoring and food qual-ity control, corrosion and wear protection.14 In order the application these films is successful,

it is essential to precisely know their elastic modulus, thickness and mass density In general, the measurement of free standing ultrathin film properties is highly complicated,5 therefore the films are usually deposited on the substrate materials and their properties are then evaluated from the measured quantities of the prepared substrate-film structures themselves For such structures the film elastic modulus is usually determined by for instance the scanning probe microscopy,6 bulge test,7 nanoindentation,8strain elastic instability9and resonant methods.10Whereas the film thickness can be measured by the ellipsometry,11 Raman spectrometry12 or X-ray diffraction tech-nique.13For ultrathin films, i.e film thickness of tens nm, even the material densities are can varies essentially from the bulk values available in literature In response, many sophisticated methods to measure density of ultrathin films have been also developed.13–18 Among them just the resonant

a Corresponding Author email address: stachiv@fzu.cz (I Stachiv)

045005-2 Stachiv et al. AIP Advances 6, 045005 (2016)

based methods have been proven to simultaneously determine film elastic modulus and density reducing the cost and time of experiments.15–18

We must emphasis here that the substrate and deposited materials are often having different thermal expansion characteristics and also the film adhesion onto different substrate materials can vary widely.19 , 20Therefore, the residual stress is commonly present in the prepared substrate-film structures and the one either enhances or degrades performances of the prepared structure or limits its lifetime.21,22To measure residual stresses in thin films requires use of other sophisticated tech-niques such as X-ray method,23bulge test,24curvature measurement25or resonant based method.26 But, for example, the use of X-ray method is limited by the correct analysis of noncrystalline materials, while for bulge test measurement of the membrane deflection is highly complicated resulting in over/ under estimation of the evaluated residual stress On the other hand, the reso-nant based method developed by Ma et al.26is generally capable to simultaneously measure film elastic modulus and the created residual stress However, the one require precise knowledge of the substrate dimensions, film thickness and density, and is limited to only a circular membrane Noticing that the most of commonly used micro-/nanomechanical actuators and sensors are having

a rectangular cross-sectional area.27 , 28

Evidently, it is of practical importance to develop an easy accessible method enabling simul-taneous determination of the film elastic modulus, density, thickness and created residual stress

on structures with rectangular or squared cross-sectional areas In this work, we firstly show that

a general dependency of the beam resonant frequencies on the axial force derived primarily for out-plane vibrations is also valid for the in-plane ones and its slope depends just on the consid-ered vibrational mode Consequently, based on the obtained results, we found that the film elastic modulus, density and thickness can be calculated from two in- and out-plane fundamental reso-nant frequencies of beam under different intentionally applied prestress forces before and after film deposition Whereas, to determine the residual stress require measurement of two out-plane (in-plane) consecutive resonant frequencies of doubly clamped beam with deposited ultrathin film vibrating again under intentionally applied prestress force(s) Importantly, the axial prestress forces acting upon the micro-/nanobeams can be easily generated and controlled via an external electrical and magnetic fields28 – 31or even mechanically.32

Paper begins with a general theory and its application limits In the second part of paper, Sec III, method of the residual stress measurement is proposed and the effect of inaccuracies in force measurement on the extracted data is evaluated In the last part of paper, Sec.IV, we present a method capable to simultaneously determine elastic modulus, density and thickness of the polymer and solid ultrathin films

II BACKGROUND THEORY

The following model is limited to the flexural vibrations of the doubly clamped beam of either rectangular or squared cross-section and with or without deposited solid or polymer ultrathin film(s) under an arbitrary value of axial force (see in Fig.1(a)) Then, the in-plane and out-plane flexural vibrations of beam of length L, width W , thickness Tc, elastic modulus Ec and density ρc with deposited ultrathin film of thickness Tp, elastic modulus Epand density ρpunder an arbitrary value

of axial load FTare described by the following equation

Aρ

∂2u(x,t)

∂t2 + DF

∂4u(x,t)

∂x4 − FT

∂2u(x,t)

where Aρ= W(ρcTc+ ρpTp) is the mass of beam per unit length, and the flexural rigidity DF

= (W3/12)(EcTc+ EpTp) for in-plane vibrations and DF= (W/12)[EcTc + EpTp + 2EcEpTcTp(2Tc

+ 2T2+ 3TcTp)]/(EcTc+ EpTp) for out-plane one.18 Accounting for two clamped ends boundary conditions and solving Eq (1) yields spectrum of the flexural resonant frequencies

f = γ 2/(2πL2

FIG 1 a) Sketch of the considered doubly clamped beam with deposited thin layer film; (b) a general dependency of resonant frequency on tension parameter b for first two consecutive resonant frequencies.

where a= E c I c r(υ,η)

ρcWTC(1+ξη), r(υ, η) = 1 + υη and [υ2η4+ 4υη(1 + 1.5η + η2

) + 1]/(1 + υη) for in-plane and out-in-plane vibrations, respectively; η = Tp/Tc, υ= Ep/Ec, ξ= ρp/ρc, and γn2is spectrum

of the dimensionless resonant frequencies obtained as a solution of the following transcendental equation

cosh qi cos qj− 1 ± b2/(2q1q2) sinh qi sin qj= 0, (3) where q1,2= [±b2/2 + (b4/4 + γ4

)1/2

]1/2, b= FTL 2

I c E c r (ν,η) is the tension parameter,33 Ic= W3Tc/12 and WTc/12 for in-plane and for out-plane vibrations, and i = 1 (2) and j = 2 (1) stand for a tensile (compressive) axial force

It is evident from structure of Eq (3) that its solution describes a complete spectrum of the dimensionless resonant frequencies of beam under an arbitrary value of tensile or compressive axial load represented through the tension parameter b In addition, for a given axial load the partic-ular values of the dimensionless resonant frequency γ2vary essentially for in-plane and out-plane vibrations, i.e for in-plane and out-plane ones the moment of inertia Icand the coefficient r(υ, η) are different from each other Furthermore, to verify that dependency of γ2 on b is indeed valid also for in-plane vibrations, we solve Eq (1) for a large number of axial forces, beam dimensions and mechanical properties, and both in-plane and out-plane vibrations Then, results presented in Fig.1(b)for first two vibrational modes and a tensile force confirmed that dependency of γ2on b is general and valid for both in-plane and out-plane flexural oscillations

III RESIDUAL STRESS DETERMINATION

In this section, method of the residual stress measurement utilizing detection of out-plane reso-nant frequencies of doubly clamped beams is proposed and the accuracy of the one is analyzed To begin, we recall a known fact that a general dependency of γ2on b for out-plane vibrations of beam

of different cross-sectional area(s) and with or without deposited thin film(s) under an arbitrary value of FT has been already found and explained.18 , 33 , 34 It has been also shown that in case of doubly clamped micro-/nanosized beams the residual (surface) stress creates a net axial force Fres.35

Moreover, it is worth noting that most of the micro-/nanosized beam are designed with a dominated flexural rigidity.33 – 37Thus computing γn2over a large number axial forces and materials properties and with due account for|b| < 1.5, dependency of γn2on b can be accurately approximated by the polynomial function of the fourth order For first mode the first dimensionless resonant frequency

045005-4 Stachiv et al. AIP Advances 6, 045005 (2016)

can be expressed as

γ1 = a4b4+ a3b3+ a2b2+ a1b+ γB12 (4a) and, consequently, for second mode the dimensionless resonant frequency are given by

γ2 = c4b4+ c3b3+ c2b2+ c1b+ γB22 (4b) where a1= −0.0014, a2= 0.5548, a3= −0.006, a4= −0.0048, c1= −0.0005, c2= 0.7485, c3

= −0.0022, c4= −0.0039, γB12and γB22are the dimensionless resonant frequencies obtained as a solution of Eq (3) for b= 0, e.g for first mode γB12≈ 22.37 and for second one γB22≈ 61.67 The ratio between two measured consecutive resonant frequencies of beam with deposited film yields the following quartic equation

(Rc4− a4)b4+ (Rc3− a3)b3+ (Rc2− a2)b2+ (Rc1− a1)b + (RγB22−γB12) = 0, (5) where R= f1/ f2 is the ratio between measured first and second resonant frequency, b= b0[(FT

+ Fres)/r(υ, η)]0.5 and b0= L/(IcEc)0.5 As can be seen from Eq (4) and is also depicted for first two consecutive resonant frequencies in Fig.1(b)dependency of γ2on b differs essentially for first and second vibrational mode It immediately implies that the residual force created by the residual (or surface) stress due to film deposition can be unambiguously determined from two measured out-plane (in-plane) consecutive resonant frequencies of beam with film vibrating under different intentionally applied axial prestress forces Then, solving Eq (5) for two different values of FT and accounting for a physical meaning of the dimensionless resonant frequency, i.e γ2can be just a real and positive number, results in the desired expression for Fresin the following form

Fres≈(R1 FT 2− R2 FT 1)/(R2 − R1), (6) where FT 1and FT 2are two different intentionally applied prestress forces, and the coefficients R1,2

are the positive and real roots of Eq (5) obtained numerically for the variable b0/r(υ, η)0.5 and they contain just and only known values of the polynomial coefficients a and c, and two different ratios between measured resonant frequencies of beam with deposited film, i.e ratios of frequencies for two different applied prestress forces We only mention here that Eq (5) can be solved even analytically by reducing the original quartic equation, Eq (5), to the depressed quartic equation and then using a well-known Ferrari‘s solution.39However, this analytical solution is time consuming and the final solution is bulky and cumbersome, therefore the numerical solution of Eq (5) can be naturally preferable

In addition, a following very important conclusion can be drawn from Eq (6): the residual force

Fres can be determined without knowing the substrate and film dimensions, densities and elastic moduli This finding is of practical importance in design of tunable micro-/nanomechanical reso-nators,28where precise knowledge of the created residual (surface) stresses caused by film deposi-tion methods is crucial for the correct estimadeposi-tions of resonators‘ operating condideposi-tions and further performances Moreover, the accuracy of determined Fresdepends on uncertainties in the resonant frequencies and applied axial prestress forces measurements and the considered polynomial depen-dency.37,38Typical uncertainties in force measurement, i.e for microbeams of O(0.1 pN)40and for nanobeams of O(0.1 fN),41have a negligibly small impact on the resonant frequencies,42 thus the relative error in residual force determination ∆Frescaused by the sensitivity in force measurement

∆FTcan be expressed in the following way

∆Fres

Fres

= R

2

1FT 2

(

1+∆ T 2

FT 2

)

− R2

2FT 1

(

1+∆ T 1

FT 1

)

R2

1FT 2− R2

2FT 1

To illustrate that here proposed resonant method of the residual stress determination is realy prac-tical, we present in TableIdetermined achievable errors ∆Fresfor beam made of silicon (Ec= 169 GPa,

ρc= 2.33 g/cm3, L= 200 µm,W = 20 µm, andTc= 1 µm) with a) 20 nm thick AlN film (Ep= 350 GPa and ρp= 2.33 g/cm3) and b) 40 nm thick Au film (Ep= 79 GPa and ρp= 19.3 g/cm3), where the axial loads are F = 1 µN, F = 7 µN, F = 13 µN and the uncertainties in force measurements are of

TABLE I Error in residual force determination for i) silicon beam with 20 nm AlN film and ii) silicon beam with 40 nm Au film The applied axial forces are F T 1 = 7 µN and F T 2 = 13 µN, respectively.

∆F T 1 ,

nN

∆F T 2 , nN

∆F res (AlN) nN

∆F res (Au) nN

piconewtons, i.e one order higher than the commonly achievable force sensitivity of microbeams Re-sults given in TableIreveal that the achievable error in residual force measurement is of the same order

of magnitude as the uncertainties in applied prestress forces Consequently, we can easily conclude that the common uncertainties in force measurement have negligible effect on the present method of stress measurement Consequently, errors in applied prestress forces can be simply neglected without

affecting accuracy of the extracted residual stress values

IV FILM ELASTIC MODULUS, DENSITY AND THICKNESS DETERMINATION

Here, the resonant method of ultrathin film elastic modulus, density and thickness determina-tion is proposed and the expressions enabling fast and yet accurate calculadetermina-tions of ultrathin film properties from the measured fundamental resonant frequencies of micro-/nanobeam are derived, and the achievable accuracies of the determined properties are obtained and analyzed As mentioned

in previous section, Sec.III, for|b| < 1.5 the fundamental resonant frequencies of beam with and without deposited ultrathin film under applied axial prestress force can be estimated from Eq (4a) Hence, with help of Eq (4a) the ratio between two fundamental resonant frequencies of beam with film under different applied prestress forces can be written in the following way

fp1

fp2 =a4b

4 p1+ a3b3p1+ a2b2p1+ a1bp1+ γ2

B1

a4b4p1+ a3b3p1+ a2b2p1+ a1bp1+ γ2

B1

where coefficiens a1,2,3,4 are given in Eq (4), subsripts 1 and 2 stand for two different inten-tially axial prestress forces, i.e.FT 1 and FT 2,43 and bp= b0[(FT+ Fres)/r(υ, η)]0.5 Importantly, in comparison to the residual stress measurement, for film properties determination the second order polynomial dependency of γ2on b still enables relatively accurate results, therefore the ratio of two fundamental resonant frequencies of beam with film can be further expressed by

fp1

fp2 = α2b

2 p1+ α1bp1+ γ2

B1

α2b2p2+ α1bp2+ γ2

B1

(9a) and, similarly, for beam without film the frequency ratio reads

fc1

fc2

=α2b2c1+ α1bc1+ γ2

B1

α2b2c2+ α1bc2+ γ2

B1

where coefficients α1≈ 0.49885, α2≈ 0.05736 and bc= b0FT0.5 From Eqs (9a) and (9b), we conclude that frequency ratios of beam with and without film differ from each other only through

r(υ, η) Solving Eqs (9a) and (9b) for bcand bp, and, then, accounting for ratio of bc/bpyields the expression for r(υ, η)

r(υ, η) =*

−G2+G22− 4G1G3

−G5+G2

5− 4G4G6

+ /

2

G24

G2 1

045005-6 Stachiv et al. AIP Advances 6, 045005 (2016)

where G1= α2(KcFT 2− FT 1), G2= α1(KcFT 20.5− FT 10.5

], G3= γB12(Kc− 1), G4= α2[(KpFT 2− FT 1) + (Kp− 1)Fres], G5= α1[Kp(FT 2+ Fres)0.5−(FT 1+ Fres)0.5

], G6= γB12(Kp− 1), Kc= fc1/ fc2 and

Kp= fp1/ fp2

Notably, the coefficent r(υ, η) depends on the film elastic modulus and its thickness; two film properties that are needed to be determined But from Eq (1), Fig 1(b) and discussion given

in Sec II, we show that in-plane and out-plane resonant frequencies differ from each other just through the moment of inertia Icand coefficient r(υ, η) Hence, by measuring both the in-plane and out-plane resonances, the effect of elastic modulus and thickness on r(υ, η) can be easily disen-tangle Then, substituting the explicit expressions for r(υ, η) given in Eq (1) and rearranging terms

in Eq (10) gives the seek equation for calculation of the film thickness

η = −3

Sin+ 3+

 9 (Sin+ 3)2−(4Sin− SinSout− 3)

(Sin− 1)(Sin+ 3) (11a) and, coresspondingly, for the film elastic modulus we obtain

where S= *

,

−G2+ 

G 2 −4G1G3

−G5+ 

G 2 −4G4G6

+

-2

G2

G2, subscripts in and out stand for in-plane and out-plane measured resonant frequencies, respectively Film mass density is calculated from the measured ratio of beam with and without film

ξ = [( fc/ fp)2r(υ, η) − 1]/η (11c)

It is important to note that the errors in evaluated thickness, elastic modulus and mass den-sity of deposited ultrathin film depend on the uncertainties in beam thickness measurement As a result, Eqs (11a) – (11c) can be also used to derive sensitivity of the calculated film properties to inaccuracies in beam thickness We suppose the small uncertainty in thickness measurement ∆Tc

including the corresponding errors in ∆r(υ, η) and ∆S Then by means of perturbation technique the sensitivity in thickness ∆Tp, elastic modulus ∆Epand density ∆ ρpdeterminations can be obtained in the following way

∆Tp/Tp≈(Sin+ 3)(1+∆Tc

Tc

) 

Sin

(

1+∆Sin

Sin

) + 3

−1

∆Ep/Ep=



Sin

( ∆Sin

Sin +∆Tc

Tc

)

−∆Tc

Tc



∆ρp/ρp= ( fc/ fp)2r(υ, η)[∆r(υ, η)/r(υ, η) + ∆Tc/Tc]/[( fc/ fp)2r(υ, η) − 1] (12c)

To illustrate the application potential of present method of film elastic modulus, density and thickness measurement, we suppose silicon beam used previously for the residual stress determina-tion under Fres= 1 µN, FT 1= 6 µN and FT 2= 12 µN, and SiO2beam (Ec= 75 GPa, ρc= 2.2 g/cm3,

L= 200 µm, W = 20 µm, and Tc= 1 µm) under Fres= 1 µN, FT 1= 0.5 µN and FT 2= 5 µN with deposited 20 nm thick AlN and 40 nm thick Au thin films used previously for Fresdetermination

TABLE II Calculated elastic moduli, densities and thickness of AlN and Au thin films deposited on doubly clamped beam made of silicon and SiO 2 with accounting for the uncertainties in thickness measurement of 1 nm, i.e 0.1 %.

Configuration r in (υ, η)/r out (υ, η)

E p /∆E p , GPa

ρ p /∆ρ p ,

g /m 3

T p /∆T p , Nm

The error in substrate thickness measurement ∆Tc= 1 nm, i.e this error is one order of magnitude more accurate than the commonly achievable resolution in thickness measurement.44 , 45For readers convenience, the calculated values of film properties and the corresponding errors are summarized

in Table II As can be seen from Table IIthe calculated properties of both ultrathin films are in

a good with the expected data In addition, the calculation of film elastic modulus, density and thickness by Eq (11) can be performed without measuring beam length and width And, noticing only that the accuracy of present method can be essentially improved by employing the fourth order polynomial dependency of γ2on b, whereas the method of film properties calculations remind unchanged

V CONCLUSIONS

In this work, we show that the general dependency of the beam resonant frequency spec-trum on tension parameter derived previously for out-plane vibrations is as well valid for in-plane ones and varies only with considered vibrational mode Then, we used these findings and pro-posed method capable to simultaneously determine the elastic modulus, density, thickness and the corresponding residual stress of the solid and polymer ultrathin film deposited on the doubly clamped micro-/nanobeam In present method the created residual (surface) stress is calculated from two measured consecutive resonant frequencies of beam with film under different intentionally applied prestress forces, while the other film properties are determined from two tuned in-plane and out-plane fundamental resonant frequencies of beam with and without deposited film Accuracies

of the extracted values and the estimated properties of ultrathin films are in a good agreement with the expected data We also show that the typical errors in force measurement have negligible impact

on the residual force and film properties measurements Our results demonstrate the application potential of resonant nanomechanics in a non-destructive material testing Here proposed method

of film properties measurement can help not only to significantly reduce the required time and cost

of experiments but also help to explain the recently observed anomalous dynamic of nanobeams caused by the stress induced molecule adsorption

This work was supported by the Grant Agency of Czech Republic, under GACR 15-13174J and by the Ministry of Science and Technology, Taiwan, under MOST 104-2218-E-151-002 and 103-2221-E-151-001-MY3

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