Depth-sensing indentation modeling for determination of Elastic modulus of thin films

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Depth-sensing indentation modeling for determination of Elasticmodulus of thin films

A. Tricoteaux a,b,*, G. Duarte b, D. Chicot b, E. Le Bourhis c, E. Bemporad d, J. Lesage b

a Laboratoire des Matériaux et Procédés, LMP, LMP – Université de Valenciennes et du Hainaut Cambrésis, Z.I. du Champ de l’Abbesse 59 600 Maubeuge, Franceb Laboratoire de Mécanique de Lille, LML, UMR 8107, U.S.T. Lille, IUT A GMP, BP 179, Rue de la Recherche, 59 653 Villeneuve d’Ascq, Francec Laboratoire PHYMAT, UMR 6630, Université de Poitiers, SP2MI-Téléport 2-Bd M&P Curie-BP 30179, 86 962 Futuroscope-Chasseneuil Cedex, Franced University of Rome ‘‘ROMA TRE”, Mechanical and Industrial Engineering Department, Via Vasca Navale 79, 00146 Rome, Italy

a r t i c l e i n f o

Article history:Received 5 December 2008Received in revised form 10 September 2009

Keywords:Depth-sensing indentationNanoindentationThin filmsElastic modulusModelTiCN

a b s t r a c t

There are various methods to address the problem of determining the hardness of thinfilms when the substrate is involved in the deformation process produced during conven-tional indentation tests. For the determination of the elastic modulus using depth-sensingindentation methods, the problem is more complex due to the deformation of the equip-ment that comes in addition to the effect of the substrate.

In the paper we discuss the use of Oliver and Pharr’s method to take into account thedeformation of the equipment for the measurement of the elastic modulus of TiCN thinfilms.

For micro-indentation tests we proposed a new model to precise the effect of the sub-strate. The elastic moduli that are calculated are in very good agreement to those foundin literature. For nano-indentation tests it is necessary to correct the data to take intoaccount the shape of the indenter tip. We show that this correction, proposed at the originfor massive materials, is not able to explain the discrepancies between the calculated val-ues and those coming from the literature for the elastic modulus of thin films.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

During the last decades, numerous works have been con-ducted to develop new surface modification process in orderto increase the resistance of mechanical components to sur-face damage. Among them, physical and chemical vapordeposition provide very hard films of thickness of the orderof micrometer. In addition to the hardness, Leyland and Mat-thews (2000) and Batista et al. (2003) have shown that theelastic modulus of the film is an important parameter to con-sider when studying the performance of such coated mate-rials. Mechanical properties of thin films cannot be

determined using conventional tensile test. A popular solu-tion is to deform the coating at a very small scale usingdepth-sensing indentation. For the thicker films, the methodof Oliver and Pharr (1992) proposed to analyze nano-inden-tation experiments, allows a direct determination of theelastic modulus of the film. For the thinner films, this meth-od cannot be applied and direct determination is not possi-ble due to the interaction with the substrate. Chudoba et al.(2002) and Cleymand et al. (2005) have observed that theinfluence of the substrate begins to be noticeable for inden-tation depth as low as 1% of the film thickness. This is muchlower than the influence observed which begins to benoticeable for indentation depths near 10% of the film thick-ness depending on the hardness and nature of the film (Sunet al., 1995; Cai and Bangert, 1995). Following these observa-tion, it is clear that any measurement performed either inthe nano or the micro-indentation range, would involvethe substrate for films of thickness near and above 1 lm that

0167-6636/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmat.2009.11.016

* Corresponding author. Address: Laboratoire des Matériaux et Procé-dés, LMP, LMP – Université de Valenciennes et du Hainaut Cambrésis, Z.I.du Champ de l’Abbesse, 59 600 Maubeuge, France.

E-mail address: [email protected] (A. Trico-teaux).

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are the range of thickness used for mechanical applications.Models to separate the contributions of film substrate to theglobal deformation were proposed recently for the analysisof load–depth indentation curves. These models (Doernerand Nix, 1986; Mencik et al., 1997; Gao et al., 1992; Antuneset al. 2007) are based on the best fit analysis of the load–depth curve by an appropriate function. These very simplefunctions cannot describe properly the two asymptotic ten-dencies that should be observed for the very low loads forwhich the elastic modulus tends toward that of the filmand for the higher loads for which the modulus tends towardthat of the substrate. In addition, from the experimentalpoint of view, it is important to keep in mind that indenta-tion depth measured during the indentation test is due tothe deformation of the specimen itself and to the deflectionof the whole instrument. In a recent paper, Fischer-Cripps(2006) considered that the compliance of the equipmentframe involves also the specimen mounting. As a conse-quence, the compliance is not a constant value but have tobe estimated for each specimen when applying Oliver andParr’s (1992) method.

The objective of the present work is to propose a newmodel based on two requirements

(1) to represent the two asymptotic tendencies for thefilm and the substrate;

(2) to propose a general method for the evaluation ofthe compliance of the global system.

The model will be applied to the mechanical character-ization of TiCN thin films, of thickness near to 2 lm, depos-ited on a steel substrate.

2. Background theory

2.1. Measurement on a massive material

For depth-sensing indentation analysis, Oliver andPharr (1992) proposed to determine a ‘‘reduced elasticmodulus” as a function of the indentation parameters de-duced from the unloading part of the load–depth curve(Fig. 1):

ER ¼1b

ffiffiffiffipp

21ffiffiffiAp 1

Cð1Þ

where ER is the reduced modulus, b a correction factor, Athe contact area and C is the compliance.

The reduced modulus, ER, depends on the elastic modu-lus and Poisson’s coefficient of the specimen and those ofthe indenter as follows:

1ER¼ 1� m2

Eþ 1� m2

i

Eið2Þ

For a diamond indenter, the elastic modulus, Ei, and thePoisson’s coefficient, mi, were found equal to 1140 GPaand 0.07, respectively (Field and Telling, 1999).

In relation (1), the correction factor b is linked to the in-denter geometry. King (1987) proposed a value b = 1.012for a Vickers indenter, whereas Dao et al. (2001) proposeda value of 1.07. Recently, a three dimension simulation of

the Vickers indentation lead to b = 1.05, a value almostindependent on the material (Antunes et al., 2006). Thecontact area, A, is calculated from the experimental contactdepth hc:

A ¼ 24;5h2c ð3Þ

where

hc ¼ hmax � ePmax

Sð4Þ

Coefficient e is a factor depending on the geometry of theindenter shape. e is equal to 1 for a flat punch indenter,0.72 for a conical indenter and 0.75 for a spherical/parabo-loïd indenter (Oliver and Pharr, 1992). Compliance, C, inrelation (1) corresponds to the inverse of the contact stiff-ness, S, and is calculated from the slope of the unloadingcurve at the peak load, i.e. at the maximum indentationdepth:

C ¼ 1S¼ dh

dP

� �h¼hmax

ð5Þ

Oliver and Pharr also suggested the use of a power law todescribe the unloading curve as follows:

P ¼ Kðh� hf Þm ð6Þ

where K and m are material constants determined by bestfit analysis.

In order to avoid possible artifacts at the two extremi-ties of the unloading curve, i.e. at the beginning of theunloading at the maximum load and just before the com-plete unloading; only data ranging between 40% and 98%of the maximum load were used.

By combining relation (5) and (6), we can write:

C ¼ 1S¼ 1

mKðhmax � hf Þm�1 ð7Þ

Using this method, Hay et al. (1999) found out that Oliverand Pharr’s method overestimates the elastic modulus of a

hc

Unloading dh

dP

S

hmaxhf

Depth, h

Loading

Loa

d, P

PmaxCreep

Initial surface Surface profile

after load removalIndenter

hf

hchmax

Pmax

Surface profile under load

(b)

(a)

Fig. 1. Schematic representation of a load–depth curve.

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massive material. Following Hay et al., this overestimationis due to radial displacements that occur inside the contactcircle when Sneddon solution is applied to the indentationof a half-space by a rigid cone. As an improvement, theauthors introduce a correction factor c function of the solePoisson’s coefficient, to be used in Eq. (1):

c ¼ pp4 þ 0;1548 1�2m

4ð1�mÞ cot Wp2 � 0;8312 1�2m

4ð1�mÞ cot Wð8Þ

where W represents the half-angle of a conical indenter.When using a pyramidal indenter, W becomes the half-an-gle of an equivalent conical indenter which will producethe same contact area by applying the same indentationload (Fischer-Cripps, 2004). As a consequence for a Vickersindenter having a square based pyramid, W is equal to70.32�.

Relation (1) is consequently modified as follow:

ER ¼1bc

ffiffiffiffipp

21ffiffiffiAp 1

Cð9Þ

A formula for compliance, C, is obtained by combining rela-tions (3) and (9):

C ¼ 1bc

ffiffiffiffipp

21ffiffiffiffiffiffiffiffiffiffiffi

24;5p 1

ER

1hc

ð10Þ

2.2. Measurement on a coated material

We have seen that contribution of substrate to theindentation measurement is a function of the indentationload which is applied. For the lowest loads, the deforma-tion of the substrate is very low and the system behavesglobally like the film whereas for higher loads the systembehaves more like the substrate. When the indentationdepth increases, the apparent elastic modulus thenchanges gradually from the two limits: EF (for low loads)to ES (for higher loads). In the recent studies, models devel-oped to separate the two contributions were based on thedefinition of a ‘‘composite modulus, EC” expressed as afunction of the substrate modulus, ES and the film modu-lus, EF. In order to simplify the expressions that are goingto be presented, we will use the following relations inthe rest of the text:

1E�C¼ 1� m2

C

EC;

1E�S¼ 1� m2

S

ESand

1E�F¼ 1� m2

F

EFð11Þ

where mC, mS and mF are Poisson’s coefficients for composite,substrate and film materials, respectively.As assembled inTable 1, various relations developed by some authors rep-resent the composite modulus as a combination of those of

the substrate and film in relation to the contact indenta-tion depth, hC, the film thickness, t, and various empiricalparameters. It should be noted that in the models, E�C is ex-pressed without considering any possible variation of thecompliance Cf of the testing equipment (i.e. the frame).We will see in the following that Cf is a very importantparameter to consider for the analysis of the results. InTable 1, U is the weight function coming from Gao’s model:

U ¼ 2p

arctanta

� �

þ 12pð1� mÞ ð1� 2mÞ t

aln 1þ a

t

� �2� �

�at

1þ ðat Þ2

" #

ð12Þ

where a is the contact radius for a flat punch indenter. Ifthe usual pyramidal indenters, such as Vickers or Berko-vich, are assumed equivalent to conical indenter havingthe half-angle W, the contact radius can be deduced fromthe contact indentation depth, hc, as follow: a ¼ hc tan W.

3. Material and experiments

A micro-hardness Tester CSM 2-107 equipped with aVickers diamond indenter was used for the micro-indenta-tion experiments. The tests were performed at maximumloads chosen within the range 50–10,000 mN. Loadingand unloading rates (in mN/min) were set up at twicethe maximum applied load. A dwell time of 15 s was im-posed according to the standard CSM test procedure.Fig. 2 shows an example of loading-unloading curves ob-tained for a TiCN film of 2.19 lm thick.

For nanoindentation experiments, the samples were in-dented at room temperature with a Berkovich diamondpyramid indenter using a CSM nanohardness tester. Thetests were performed in air using the force-control modeof the machine. The loading-unloading procedure was asfollows: loading to the maximum load in 30 s, dwell-loadat peak load during 15 s, unloading in 30 s. The peak loadvaries between 5 and 300 mN. The calibration proceduresuggested by Oliver and Pharr (1992) is used to correctthe data for the load-frame compliance of the apparatus.b is equal to 1.034 for a Berkovich indenter andW = 70.32� (Hay and Pharr, 2000) and A ¼ 24;56h2

c for aperfect Berkovich tip.

4. Application of the models for the determination ofTiCN films elastic modulus

As known from the ‘‘rule of 1%”, it is not possible todetermine 1/SF directly from the micro-indentation data

Table 1Different weight functions (WF) available in literature.

Weight function 1 Doerner and Nix (1986) Weight function 2 Mencik et al. (1997) Weight function 3 Mencik et al. (1997)

j1=E�c�1=E�f jj1=E�f �1=E�s j

¼ eð�a1thÞ

jE�c�E�s jjE�f �E�s j

¼ eð�a2htÞ j1=E�c�1=E�s j

j1=E�f �1=E�s j¼ eð�a3

htÞ

Weight function 4 Antunes et al. (2007) Weight function 5 Gao et al. (1992) Weight function 6 Antunes et al. (2007)jE�c�E�f jjE�f �E�s j

¼ eð�a4thÞ

jE�c�E�s jjE�f �E�s j

¼ U j1=E�c�1=E�s jj1=E�f �1=E�s j

¼ U

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because, even for the lowest applied loads, the indentationdepth is higher than 1% of the film thickness and conse-quently, the substrate participates to the measured defor-mation. For, the nanoindentation measurements, theminimum indentation depths recorded were equal to2.1% of the film thickness. Direct determination of the elas-tic modulus is impossible in this case too. The differentmodels presented in Table 1 were applied to micro- andnanoindentation measurements performed on the threePACVD TiCN specimens named TiCN-1, TiCN-2 and TiCN-3 in the following. As-calculated elastic modulus and filmthickness are collected in Table 2 for the micro- and nano-indentation tests.

For micro-indentation results, depending on the model,a large scatter of the values, from 450 to 1900 GPa, wasfound for the Young modulus.

Since it provides an average value equal to 101 ± 27 GPafor the substrate that is lower that the 200 GPa expected va-lue, or it is not able to provide values for the films, models 5and 6 will not be considered in the analysis of the results.

In these conditions, models 1–4 give 643 ± 65 GPa,603 ± 84 GPa, 826 ± 112 GPa for TiCN-1, TiCN-2 and TiCN-3, respectively. Except for the TiCN-3 specimen, these val-ues are in accordance to results published by Karlssonet al., 2000 (TiCxN1�x films obtained by arc evaporationprocess wherex ¼ C

N), i.e. 580–610 GPa.For nanoindentation measurements, models 5 and 6

also provide very low 98 and 136 GPa and very high values412 and 414 GPa for the elastic modulus of the substrate.These models are not considered in the results.

Nevertheless, a large scatter of the mean values calcu-lated for Es and Ef is still observed, 345 ± 18, 560 ± 43,

0 1000 2000 3000 4000 5000 6000 7000 80000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

load

(m

N)

indentation depth (nm)

Fig. 2. Load–depth curves obtained with various peak loads applied to TiCN-1 specimen.

Table 2Application of the different models presented in Table 1 for micro and nanoindentation tests (WF*: weight function).

WF* TiCN-1 t = 2190 nm TiCN-2 t = 2530 nm TiCN-3 t = 2540 nm

EF (GPa) ES GPa) R2 EF (GPa) ES GPa) R2 EF (GPa) ES GPa) R2

MICRO indentation experiments1 528 204 0.99 453 130 0.96 619 178 0.992 657 241 0.98 640 181 0.93 872 225 0.983 661 220 0.98 610 140 0.98 955 196 0.994 728 200 0.99 709 151 0.96 858 160 0.985 709 136 0.99 730 96 0.94 841 71 0.986 NC 194 0.97 1920 207 0.94 NC 175 0.96

NANO indentation experiments1 310 143 0.95 599 271 0.95 770 300 0.972 361 175 0.95 610 295 0.92 870 338 0.953 356 137 0.98 489 280 0.92 754 318 0.964 355 162 0.95 541 264 0.95 875 288 0.975 319 98 0.99 371 194 0.98 385 412 0.846 369 136 0.97 384 217 0.98 369 414 0.96

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817 ± 50 GPa for Ef, and 154 ± 13, 277 ± 10, 311 ± 17 for Es

respectively.The main reasons for such a scatter of the results can be

put forward.First of all the deformations of the experimental setup

and the mounting device apart from that of the specimenitself can contribute to the total displacement measuredduring the indentation process. These deformations thatwill appear in the rest of the text under the name of ‘‘framecompliance”.

The second reason comes from the form of the functionsthemselves. A powerful function should be able to repre-sent the two tendencies corresponding to loads very low(film tendency), or very high (substrate tendency) i.e. amodel presenting two asymptotic limits for the two ten-dencies. This is not the case for the models proposed in lit-erature and therefore they cannot describe the elasticmodulus variations with indentation load between thelimits.

The third reason is related to the real area of contact be-tween the indenter and the surface of the sample. Indeed,the rounded tip of the indenter can influence the elasto-plastic deformation zone around the indent, especially atthe beginning of the indentation test corresponding to verysmall indentation depths. But in practice, this phenomenonis negligible for macro and micro-indentation but can besignificant in the nanoindentation range.

5. The new model

According to the preceding remarks, a relevant modelshould take into account the compliance of the equipmentframe as well as the two asymptotic film and substratetendencies that constitute boundaries for the apparentYoung modulus. In addition for nanoindentation results,the rounded tip of the indenter should be taken intoaccount.

Let us examine first what have proposed Oliver andPharr (1992) to deal with the compliance of the frame.

5.1. Compliance

The authors assumed that the global apparent compli-ance, associated to the whole deformation, would be thesum of two distinct terms in respect to the frame Cf, andto the specimen, Cs:

C ¼ Cs þ Cf ð13Þ

Relation (10) may be rewritten as:

C ¼ Cf þ1bc

ffiffiffiffipp

21ffiffiffiffiffiffiffiffiffiffiffi

24;5p 1

ER

1hc

ð14Þ

where Cf is the ordinate at 1/hc equaling zero in a graphrepresenting the total compliance as a function of the in-verse of the contact depth.

5.2. Influence of a film

For a massive material, a representation of the result ofa depth-sensing indentation test should be a unique

straight line of slope proportional to 1/ER. The same inden-tation performed on a thin film would give a similarstraight line if there was no influence of the substrate. Aswe have seen before, it is not the case and as a conse-quence, the representation of the relation C ¼ f 1

hc

� �would

exhibit two asymptotic limits corresponding to the straightlines associated to the substrate itself and to a theoreticalstanding alone film as it is shown schematically in Fig. 3for a hard film on a soft substrate.Very often experimen-tally, it is not possible to apply sufficiently low indentationloads to render negligible contribution of the substrate andthe resulting C � 1

hcdiagram have the traditional S shape as

presented in Fig. 3 but, depending on the load range, pres-ent only a part of it as shown of Fig. 4 for micro or nanoin-dentation performed on our TiCN samples.It is clear that ananalytic model susceptible to represent this behaviorshould consider the two limits. One possible model is thatof Korsunsky et al. (1998) who consider two asymptoticlimits for the representation of the composite hardness ofa thin film in function of the normalized indentation depththat we take as hc

t :

1

1þ k � hct

� �2 ð15Þ

Models presented in literature (Tricoteaux et al., 2007) ex-press the apparent (or composite) hardness as a linearadditive law function of both substrate and film hardness:

H ¼ aHf þ bHs with aþ b ¼ 1 ð16Þ

or H ¼ Hs þ aðHf � HsÞ ð17Þ

where a is a function of indentation depth.Transposed to the apparent compliance of the whole

system, this relation can be written:

1SC¼ 1

SSþ a

1SF� 1

SS

� �ð18Þ

By combining relation (14), (15), and (18) and replacing Hby 1

SC, HS by 1

SSand HF by 1

SF, one obtains a new model for the

determination of the Young modulus of a film:

RSEslope

1α

RFEslope

1α

Film behaviour

Substrate behaviour (1/SS)

Cf Influence of the No influence of the

substrate

1/hc

1/S = dh/dP

Fig. 3. Shematic representation of the contact stiffness variation for thecoated system (hard film on soft substrate).

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1SC¼ 1

hc

1

1þ k0h2c

!1

2b

ffiffiffiffiffiffiffiffiffiffip

24:5

r� �1

cFERF� 1

cSERS

� �

þ 12b

ffiffiffiffiffiffiffiffiffiffip

24:5

r1

cSERS

� �1hcþ Cf ð19Þ

where k0 ¼ k=t2;1

ERF¼ 1

E�Fþ 1

E�i;

1ERS¼ 1

E�Sþ 1

E�iand

1E�i¼ 1� m2

i

Ei

It is noticeable this model can be applied even when theYoung modulus of the substrate is unknown.

Relation (19) can be presented as follows:

1SC¼ P0 þ

P1

hcþ P2

hcð1þ P3h2c Þ

ð20Þ

with P0 ¼ Cf ; P1 ¼1

2b

ffiffiffiffiffiffiffiffiffiffiffip

24;5

r1

cSERS;

P2 ¼1

2b

ffiffiffiffiffiffiffiffiffiffiffip

24;5

r� �1

cF ERF� 1

cSERS

� �; P3 ¼ k0

P1 depends only on the Young modulus of the substratewhereas P2 depends on both substrate and film Youngmoduli.

6. Results and discussion

6.1. Application to micro-indentation experiments

Experimental results presented in Fig. 4a show clearlythat the experimental curve does not intercept the axes atthe origin of 1/Sc � 1/hc diagram. This is the manifestation

0 500 1000 1500 2000 2500 3000 3500 4000 45000.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

0.0022

TiCN-1 (t = 2190 nm)

TiCN-2 (t = 2530 nm)

TiCN-3 (t = 2540 nm)

(a) microindentation experiments

dh/d

P, h

in m

m a

nd P

in N

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 200000.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

TiCN-1 (t = 2190 nm)

TiCN-2 (t = 2530 nm)

TiCN-3 (t = 2540 nm)

(b) nanoindentation experiments

dh/d

P, h

in m

m a

nd P

in N

1/hc, hc in mm

1/hc, hc in mm

Fig. 4. Variation of the contact stiffness as a function of 1/hc.

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of the deformation of the frame during the test. As a matterof fact, the intercept is found to be near 1 � 10�5 mm N�1 for1/hc = 0 in accordance to previous results obtained by Chicotand Mercier (2007) in our laboratory on a standard fused sil-ica specimen (Cf = 2.8 � 10�5 mm N�1) using the sameequipment.

Application of the model developed in relation (19) tothe three TiCN films shows a very good fit to the experi-mental points with an excellent correlation coefficient R2

very near 1 (Fig. 5). Also, the calculated values of the filmsYoung modulus collected in Table 3 (461–681 GPa) are of

the same order of magnitude than the ones published inliterature for TiCN coatings.

6.2. Application to nanoindentation experiments

Nanoindentation measurements are very sensitive tothe shape of the indenter tip. A calculation of the real con-tact area A, has to take into account the tip rounding. Tro-yon and Huang (2006) proposed to model a Berkovichindenter by a cone presenting a tip rounding (Fig. 6) allow-ing the contact area to be expressed as:

A ¼ p tan2 Wðhc þ hbÞ2 ¼ 24:56ðhc þ hbÞ2 ð21Þ

where hb corresponds to a truncation length,

hb ¼ R1

sin w� 1

� �ð22Þ

On the other hand as indicated in Fig. 6, the distance D canbe calculated by using the following equation:

D ¼ Rð1� sin WÞ ð23Þ

where, as for the Vickers indenter, W is equal to 70.32� forthe Berkovich indenter.

Relation (21) will be used in the following to apply themodel. For the present study, we performed experimentsat low loads on a fused silica reference sample to estimatethe radius, R, from the elastic Hertzian contact theory. Thetip radius was found equal to 350 nm and thus, hb equal to22 nm and D equal to 20.4 nm.

In Fig. 4b, presenting the diagram dh/dP versus 1/hc, onecan see that the deformation of the frame depends on thespecimen: higher the thickness, lower the values of dh/dP.Compliance of the frame (Cf) as represented by parameterP0 will then be considered as variable in the fitting proce-dure. The resulting curves and correlation coefficients R2

as presented in Fig. 7 show a very good fit between theexperimental and calculated values from the model.Although this pretty good accordance, the Young modulusvalues that are deduced from the analysis are too low if com-pared to values obtained by micro-indentation and those

0.000 0.002 0.004 0.006 0.008 0.0100.0000

0.0005

0.0010

0.0015

0.0020

0.0025

(a) TiCN-1 R2=0.99937

1/S,

in m

m/N

hc, in mm

0.000 0.002 0.004 0.006 0.008 0.0100.0000

0.0005

0.0010

0.0015

0.0020

0.0025

(b) TiCN-2 R2=0.98519

1/S,

in m

m/N

hc, in mm

0.000 0.002 0.004 0.006 0.008 0.0100.0000

0.0005

0.0010

0.0015

0.0020

0.0025

(c) TiCN-3 R2=0.99849

1/S,

in m

m/N

hc, in mm

Fig. 5. Application of the new model to micro-indentation data.

Table 3Elastic modulus calculated with the model applied microindentation data.

Sample EF (GPa) ES (GPa) R2

TiCN-1 484 232 0.999TiCN-2 461 215 0.985TiCN-3 681 227 0.998

Δhb

R

Ψ

Fig. 6. Schematic representation of a non ideal indenter tip.

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published in the literature (Bull et al. 2003; Karlsson et al.,2000). Indeed, calculated values for the Young modulusare equal to 274 GPa, 326 GPa and 390 GPa for TiCN1, TiCN2and TiCN3 specimens, respectively, whereas Karlsson et al.(2000) obtained values ranging between 580 GPa and610 GPa for TiCN films deposited by arc evaporationprocess.

Two main hypotheses can be proposed to explain theseresults:

1. For very low indentation depth on rough materials, andin particular for a spherical indenter, some authors

show that ER is underestimated because the contactradius increases with roughness (Fischer-Cripps, 2004;Jiang et al., 2008). In the present study, the indentertip radius is equal to 350 nm and can be assimilatedto a spherical indenter for very low indentation depths.Moreover, Fang et al. (2004) analyzed surface topogra-phies of PECVD TiCN films and measured rms (root-mean-square) roughness equal to 23.8 nm. This valuehas the same order of magnitude as the tip roundedlength D measured with own Berkovich indenter.Therefore the roughness of the tested TiCN films prob-ably has a significant effect on the measured Youngmodulus.

2. In the model, coefficients c and b were considered asconstants but this is not true as pointed out in a recentwork by Troyon and Huang (2006). To take into accounttheir variations with the indentation depth, the authorsproposed to modify c, b by a coefficient a that dependson the size of the rounded indenter and on the mechan-ical properties of the tested material. Anyway, Abbeset al. (2008) showed that this correction a is particu-larly significant for very low indentation depths. More-over, the problem is more complex because the testedmaterial is a composite material, and the coefficient ashould depend on both film and substrate materialproperties.

7. Conclusions

A new model is proposed to calculate the elastic modu-lus of thin films from Oliver and Pharr’s method applied todepth-sensing nano and micro-indentation experiments.This model takes into account the actual frame compliance.Moreover, the two tendencies observed at low and highloads, corresponding to film and substrate tendenciesrespectively, are introduced through a weight functionsimilarly to that of Korsunsky’s model developed for hard-ness film determination.

Applied to micro-indentation performed on variousTiCN thin films, the application of the model leads to elas-tic modulus of the films in a very good agreement withthose given in the literature. However, the model cannotbe directly used to analyze nanoindentation experimentsand some correction has to be done according to the mor-phology of the indenter, the indentation depth. In a furtherwork, special attention should also be paid also to surfacemorphology as well as the nature of the very top surfacethat could present a foreign layer (oxides, humidity, . . .)of substantial thickness compared to the very low indenta-tion depths used for nanoindentation.

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0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

(a) TiCN-1

hc, in mm

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in m

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(b) TiCN-2

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