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Materials Science and Engineering A 460–461 (2007) 95–100

Single crystal bulk material micro/nano indentation hardnesstesting by nanoindentation instrument and AFM

Liang Zhou a,b,∗, Yingxue Yao a

a School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin 150001,PR Chinab School of Material Science and Engineering, Harbin Institute of Technology, Harbin 150001,PR China

Received 4 June 2006; received in revised form 2 January 2007; accepted 11 January 2007

bstract

The micro/nano indentation hardness of single crystal aluminium and single crystal silicon are investigated. Load–depth curves can be obtainedy nanoindentation instrument, and materials indentation hardness can be calculated by Oliver–Pharr method and work of indentation method

irectly from these curves. The hardness that obtained by Oliver–Pharr method is overestimate because of material pile-up effect, and the hardnesshat obtained by work of indentation method is not very correct because of its empirical equations inaccurate. The ‘true’ hardness can be calculatedy plastic work of indentation and plastic volume that obtained by integrating fitted polynomial according to load–depth curves and atomic forceicroscopy, respectively. Comparison and analysis of the results that obtained by these methods are made. 2007 Elsevier B.V. All rights reserved.

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eywords: Nanoindentation; Hardness; Oliver–Pharr method; Work of indenta

. Introduction

As an indicator of the material’s ability against deformation,ardness has been studied for more than 100 years [1,2]. Dur-ng the last 20 years, researches have become more and morenterested in mechanical properties of smaller volumes, and the

echanical properties under micro/nano scale may differ fromhe macro scale properties due to the size effect and surfaceffect, etc. [1].

Nanoindentation as an instrumented indentation method isidely used to determine the mechanical properties of both bulk

olids and thin films. The load–depth curves including loadingnd unloading process can be obtained by nanoindentation tech-ology. It has been shown that the Oliver–Pharr method [3,4] andork-of-indentation method [5,6], which are commonly used toeasure hardness from load–depth curves.Fig. 1 shows the typical load–depth relationship curves from

anoindentation experiments, where Pmax is the peak indenta-ion load, hmax the indenter depth at peak load, and hr is the finalepth of contact impression after unloading [3].

∗ Corresponding author at: School of Mechanical and Electrical Engineering,arbin Institute of Technology, Harbin 150001,PR China.el.: +86 451 86402543/608; fax: +86 451 86413810.

E-mail address: lzhou75@163.com (L. Zhou).

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921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2007.01.029

ethod; Atomic force microscopy

The hardness usually defined as the ratio of Pmax to projectrea of hardness impression Ac (Eq. (1)), and perhaps the mostidely used method is Oliver–Pharr method [3].

op = Pmax

Ac(1)

The Oliver–Pharr method analysis procedure begins by fittinghe unloading curve to an empirical power-law relation [3,7].

= α(h − hr)m (2)

here P is the indentation load, h the indenter depth, α and mre empirically determined fitting parameters. Once the param-ters α and m are obtained by curve fitting, the initial unladingtiffness, S, can be established by differentiating Eq. (2) at theaximum depth of penetration, h = hmax. The contact depth, hc,

s estimated from the load–depth data [3,7].

c = hmax − εPmax

S(3)

here ε is a constant dependent on the indenter geometry. Forhe generally employed Berkovich indenter, it has been shownhat ε has an empirical value of 0.75 [3,7]. The project area ofardness impression function is made by fitting the Ac versus hc

96 L. Zhou, Y. Yao / Materials Science and Eng

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Fig. 1. Typical load–depth curves from nanoindentation experiments.

ata to the relationship:

c = 24.56h2c +

7∑i=0

Cih1/2i

c (4)

here Ci are constants, the lead term describes a perfecterkovich indenter, the other parameters describe deviations

rom the Berkovich geometry due to blunting at the tip [3].Work of indentation method describes indentation experi-

ents process as the use of the energy dissipated or work doneuring the indentation. The energies are based on the integral ofhe loading and unloading curves. The area under the loadingurve gives the total work Wt (gray region in Fig. 1) done duringndentation, while the elastic contribution, We (weak gray regionn Fig. 1), is given by the area under the unloading curve. Thus,he plastic work Wp (deep gray region in Fig. 1) is the differenceetween Wt and We (Fig. 1) [5,6,8].

p = Wt − We (5)

The work of indentation method to estimate material hardnessas first proposed by Stilwell and Tabor [8]. It was shown that

he conventional representation of hardness, indentation loadivided by the projected area of permanent impression, is equiv-lent to the ratio of plastic work to plastically deformed volume5,6,8]:

load, P(N)

plastic area, Ap(m2)= plastic work, Wp(J)

plastic volume, Vp(m3)(6)

The total work Wt can be obtained by integrating the load-ng curve. In general, it is found that for sharp indentation ofn elastic–plastic material the loading response is governed by= Ch2, where C is a constant and h is the penetration depth [6]

Eq. (7)).∫ hmax

∫ hmax2 Pmaxhmax

t =0

P(h) dh =0

Ch dh =3

(7)

It has been shown that the ratio of hr/hmax is equivalent tohat of Wp/Wt. Thus, the plastic work Wp can be obtained by Eq.

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ineering A 460–461 (2007) 95–100

8) [6,8].

We

Wt= 1 − Wp

Wt= 1 − hr

hmax(8)

Tuck et al. [5] suggested that the hardness could be calculatedn the basis work-of-indentation alone, and can be representedy

Wt = κP3max

9W2t

(9)

here κ is a constant equal to 0.0408 for Berkovich indenter.Alternatively, by taking the hardness to be concerned with the

lastic work of indentation, then the total work term is replacedith the plastic work and the hardness is obtained also [5] (Eq.

10)).

Wp = κP3max

9W2p

(10)

Although nanoindentation technology give significant infor-ation concerning the mechanical response to indentation andithout the need for indent image, the effects of material pile-p that often was found at the indent edges [4,6,9] can seriouslyffect the calculated values.

In addition to atomic force microscopy (AFM) high resolu-ion, three dimensional imaging capability, the use of AFM formaging residual indentations has already proved to be one of thenly methods presently available for obtaining accurate dimen-ional information from an image area of only a few microns9]. Combining Matlab soft, the real plastic deformation vol-me of micro/nano indentation can be obtained according toFM image.In this paper, the indentation hardness of single crystal alu-

inium and single crystal silicon are investigated. The hardnessf these materials is initially determined using Oliver–Pharrethod and work-of-indentation method. In order to determine

he actual deformation volume and area, the indentations areirectly measured via AFM and Matlab soft. The effect of pile-upn hardness values is investigated also.

. Experimental

Bulk materials such as single crystal silicon (10 mm ×0 mm × 1 mm) and single crystal aluminium (Ø 25 mm ×mm) were used for this experiment. Single crystal silicon sur-

ace roughness is less than 5 nm after electropolishing, and singlerystal aluminium surface roughness is less than 10 nm afterurning by diamond cutting tool in ultraprecise machine tool.

The indentation experiments were conducted with aerkovich indenter using Nano II (manufactured by MTSanoinstruments). It has load and displacement resolutions of75 nN and ±0.04 nm, respectively.In each testing run, the indenter was driven into the specimen

urface under a load gradually increased to the predeterminedepth, then unloaded after being hold at peak load for 10 s.uch a procedure repeated with different predetermined depthss 500 nm, 800 nm, 1000 nm, 1200 nm and 1500 nm for single

L. Zhou, Y. Yao / Materials Science and Engineering A 460–461 (2007) 95–100 97

Fig. 2. Load–depth curves of single crystal silicon.

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amust be less than total work, but the hardness of plastic workHWp larger than HWt. So, a simple way to use plastic work termWp replacing Wt perhaps is unreasonable in Eq. (10).

Fig. 3. Load–depth curves of single crystal aluminium.

rystal silicon and 400 nm, 500 nm, 800 nm, 1000 nm, 1500 nmnd 1700 nm for single crystal aluminium. Figs. 2 and 3 showhe load–depth curves of the single crystals silicon and singlerystal aluminium, respectively.

AFM is Dimension 3100 (manufactured by Digital Instru-ents). Its integral nonlinearity (X, Y) is 1%, XY imaging area

s 90 �m, Z range is 6 �m, and RMS vertical noise floor is0.05 nm. Figs. 4 and 5 show one of AFM images for single crys-

al silicon and single crystal aluminium at different indentationepths.

We can see that the elastic recovery appears in load–depthurves during unloading processing from Figs. 2 and 3, but thelastic recovery of single crystal aluminium is less than singlerystal silicon indicating that the deformation occurs mainly bylastic processes for single crystal aluminium. And the same

henomena can be seen from Figs. 4 and 5, the edges of singlerystal silicon bend to center because of elastic recovery, but thedges of single crystal aluminium nearly linearity except smaller

Fig. 4. AFM images of single crystal silicon.

Fig. 5. AFM images of single crystal aluminium.

ndentation. The materials pile-up appears in Figs. 4 and 5, wean see that single crystal aluminium indents pile-up higher thaningle crystal silicon.

. Results and discussion

The hardness values, Hop, HWt and HWp, of single crystalluminium and single crystal silicon that are calculated directlyrom load–depth curve using Oliver–Pharr method and work ofndentation method have been plotted as a function of maximumepth in Figs. 6 and 7, respectively.

Figs. 6 and 7 show that plastic hardness HWp is too large toccept especially in smaller depth. In addition, the plastic work

Fig. 6. Hardness–depth curves of single crystal silicon.

Fig. 7. Hardness–depth curves of single crystal aluminium.

98 L. Zhou, Y. Yao / Materials Science and Engineering A 460–461 (2007) 95–100

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Fig. 10. Plot of pile-up height with indentation depth.

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Fig. 8. Work–depth curves of single crystal silicon.

We can see that all hardness values increasing with the depthecreasing from Figs. 6 and 7. This phenomenon knows as thendentation size effect (ISE). And it is clear that the HWt < Hoprom Figs. 6 and 7, the reason is likely to be caused by thelastic contribution to the total work-of-indentation. To singlerystal aluminium, the hardness values HWt and Hop are veryimilar because that indentation deformation mainly is plasticeformation in it.

For further analysis, Tuck and Korsunsky suggesting P = Ch2

Eq. (7)) perhaps is not reasonable. If the suggestion is true,hen there is inexistent ISE and hardness is a constant, but ISEhenomenon exists in many materials micro/nano indentationxperiment and existing in this testing also.

So, we use polynomial fitting loading curves and unload-ng curves of Figs. 2 and 3, the total work and elastic workre obtained by integrating the fitted polynomial, and the plas-ic work are obtained by the difference between total work andlastic work (Eq. (5)) in this experiment. The total work, elasticork and plastic work that obtained by integrating fitted polyno-ial are named as Wt1, We1 and Wp1, respectively, in this paper

o distinguish with Wt, We and Wp that calculated by Eqs. (7)nd (8).

Figs. 8 and 9 show the work of indentation plot against thendenter depth of single crystal silicon and single crystal alu-

inium.In Figs. 8 and 9, there is large elastic work contribution to

he total work of single crystal silicon indentation process and

he elastic work contribution to the total work of single crystalluminium is very small. The work that obtained by Eqs. (7) and8) (Wt, We and Wp) not as same as the work that obtained byntegrating fitted polynomial (Wt1, We1 and Wp1), the maximum

Fig. 9. Work–depth curves of single crystal aluminium.

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Fig. 11. Work–volume curves of single crystal silicon.

ercentage of the ratio (Wt − Wt1)/Wt is nearly 20%, and theeviation must be affect hardness calculation.

The material pile-up in crystals indent edges may affect therea and volume calculation when using Oliver–Pharr methodnd work-of-indentation method. If we look at Fig. 10 we see thathe pile-up height increases with increasing penetration depthor the two crystals, and the pile-up height of single crystal alu-inium is higher than single crystal silicon at any indentation

epth because of the single crystal aluminium plastic propertyetter than single crystal silicon. The pile-up height is obtainedy AFM images also.

Combining Matlab program, we can obtain the total plasticeformation volume, including the volume of material abovend below the surface, i.e. the pile-up volume and three-sidesyramidal indent volume. The total volume, pile-up volume and

ndent volume are named as Vt, Va and Vb in this paper, respec-ively. The relationship curves of work and volume are shownn Figs. 11 and 12.

Fig. 12. Work–volume curves of single crystal aluminium.

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Fig. 13. Plots of hardness vs. depth of single crystal silicon.

In Figs. 11 and 12, we can see that the work either obtainedy Eqs. (7) and (8) or obtained by integrating fitted polynomialave a good linearity with volume, but the slopes of these fittingines are different, i.e. the hardness values are different becausef work of indentation obtained by different method.

Thus, we can obtain the hardness of crystals by work of inden-ation divided by plastic volume. When discussing the hardnessesults, the following nomenclature is used to distinguish theardness values (Hop, HWt and HWp) that have been describedn this paper. The hardness that obtained by Wt1/Vt is named

Wt1, the hardness obtained by Wp1/Vt is named HWp1, and theardness obtained by Wp1/Vb is named HWp2.

Plots of hardness, Hop, HWt, HWt1, HWp1 and HWp2 versusepth are drawn for crystals examined in the present study andre now shown in Figs. 13 and 14.

Whether the hardness calculated by Oliver–Pharr method orork-of-indentation method, and whether the work of indenta-

ion directly obtained by empirical equations or integrating fittedolynomial, the hardness values existing ISE phenomenon. Buthe hardness values that obtained by the ratio of plastic work tootal plastic deformation volume, i.e. HWp1 = Wp1/Vt variations smoother than others.

Fig. 13 shows that single crystal silicon indentation hardnessWt1 in excess of two times of HWp1, this means the large elasticork mainly contribution to the total work of indentation that

alculated by integrating fitted polynomial. Reversely, in Fig. 14,

Wt1 is almost similar with HWp1 of single crystal aluminiumecause of the deformation in indentation processing almostnly plastic deformation happening.

Fig. 14. Plots of hardness vs. depth of single crystal aluminium.

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Engineering A 460–461 (2007) 95–100 99

Further analysis, the hardness values HWp2, i.e. plastic workivided by indent plastic volume only, is very nearly the hardnessalues Hop that obtained by Oliver–Pharr method for all exam-ned crystals in present study. Hence, the Oliver–Pharr methodardness yields values have not considering the material pile-uplastic volume that would seriously affect the results. Due to thelastic contact depth is greatly underestimated in Oliver–Pharrethod because of ignoring materials pile-up, lead to overesti-ate hardness values.To HWt, the hardness values that obtained by work of inden-

ation method (Eqs. (7) and (9)) are similar with Hop of singlerystal aluminium, but similar with HWp1 of single crystal sili-on, so, the hardness value HWt is more suitable elastic–plasticaterials not plastic materials, the reason is under the under-

tanding that Hop is overestimate the indentation hardness andWp1 is nearest the ‘true’ hardness of materials.The constant κ = 0.0408 of Eq. (9) using in work of indenta-

ion method and ε = 0.75 of Eq. (3) using in Oliver–Pharr methodre dependent on the indenter geometry, but under micro/nanocale, the indenter geometry did not self-similar. So, the twoonstant will generate errors under different indentation depths.nd the project area Ac (Eq. (4)) of Oliver–Pharr method and

he hardness HWt (Eq. (9)) of work-of-indentation method arempirical equation or semi-empirical equation that will seriouslyffects by testing condition, testing instruments, irregularities ofaterial, initial state of material surface and so on. To HWp1,

he plastic work is calculated by integrating fitted polynomialccording to load–depth curves, the plastic volume includingaterial pile-up volume and indent volume are obtained by real

ndent AFM images, so, the results of HWp1 perhaps are moreccurate and believable than other results, and the effects of pile-p are eliminated because of considering the pile-up volume inalculation.

. Conclusions

Under micro/nano scale, load–depth curve can be obtainedy nanoindentation instrument, and real indent image can bebtained by AFM. Materials indentation hardness can be calcu-ated directly from load–depth curves by Oliver–Pharr methodnd work of indentation method. But the hardness values areverestimated by Oliver–Pharr method because of pile-up effect,nd the hardness values are inaccurate by work of indentationethod because of its suggestions are not completely correctly.Plastic work can be obtained integrating fitted polynomial

ased on load–depth curve and real plastic volume includingile-up volume and indent volume obtained by AFM images.he hardness values that using plastic work divided by plasticolume are more smoother than other results, and also perhapsore accurate and believable.

cknowledgements

Support for this work by the Scientific Research Founda-ion of Harbin Institute of Technology (Project: HIT.2003.23)nd Doctor Research Startup Foundation of Harbin Institute ofechnology.

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