Nanoindentation is a characterization method for determining mechanical properties of materials. Typically, nanoindentation is used for thin films and coatings, but also for bulk materials.^[1]The method is based on small-scale load applied on material surfaces with an indenter, resulting in a characteristic deformation and indentation depth of the material. The most frequently measured properties by nanoindentation are elastic modulus and hardness.^[1] Other measurable properties are residual stress, fracture toughness and creep, among others.^[2]In this article, the focus is on elastic modulus and hardness.

1. Principle

In nanoindentation, a small indenter is penetrated into solid material using a small-scale load (in the range of micronewtons). As the set load reaches its maximum value, the indenter is withdrawn from the material. As a result, an indentation is formed on the material surface (depth range in scale of nanometers).^[1] Figure 1 is a schematic representation of the indentation process.

Figure 1. Schematic illustration of nanoindentation. a) Loading: The indenter is pressed into the specimen. b) Load (or depth) reaches maximum value. c) Unloading: The indenter in withdrawn from the sample. (Figure: Topias Jussila)

During loading, the material typically first undergoes elastic deformation. As the load continues to increase, plastic deformation starts to occur, resulting in a non-linear loading curve.^[2] However, during unloading, only elastic recovery occurs as the plastic deformation formed during the loading is irreversible. Thus, an indentation (h_f) is formed with a depth smaller than the maximum indentation depth (h_max) which is depicted in figures 1-3.

There are different methods for interpretation of nanoindentation data but the most widely used one is the Oliver and Pharr method which is based on analysis of the unloading curve.^[3]In the Oliver and Pharr method, material properties are defined by the maximum load, the slope of the initial unloading curve, and the contact depth. Figure 2 is a schematic graph of a typical load-depth curve obtained from a nanoindentation measurement.

Figure 2. Load as function of indentation depth. H_r is residual depth, h_p is plastic depth, h_c is contact depth and h_max is maximum depth. (Figure: Topias Jussila)

Figure 3 provides a closer look at the interaction between the sample and the indenter. The variable h_max is the maximum depth of the measurement at the maximum load. The variable h_r is residual depth after unloading, which is smaller than h_max due to elastic recovery. The contact depth h_c is defined by the contact area between the indenter-sample interface, and it is smaller than h_max due to elastic strain at the edges of indenter-sample interface (h_s).^[1]

Figure 3. Closer look to the different depths of the indentation. (Figure: Topias Jussila)

In the initial unloading curve (Figure 2), it is assumed that the contact area between the indenter tip and the material is constant.^[3] In practice, the maximum load is held constant for a while in order to minimize the effect of creep.^[1] Creep may lead to negative contact stiffness as the indentation depth continues to increase during initial unloading.^[1] Consequently, contact stiffness is defined as the slope of the initial unloading curve which can be expressed with h_max and h_f by the power law relationship.^[3]

Where P is load, h is indentation depth, h_max is the maximum depth, h_f is the residual depth and B and m are fitting parameters.

The contact area depends on the indentation depth and the indenter geometry. There are several indenter shapes such as pyramidal, spherical and cylindrical.^[2] The most commonly used indenter is the pyramidal-shaped Berkovich tip with a typical radius of 10—100 nm.^[2] The contact area for the Berkovich tip can be approximated as follows.

Where h_c is the contact depth. In practice, the accurate tip geometry must always be calibrated as the tip typically does not possess its perfect shape due to imperfections formed over time.^[3] The calibration is done by using a material with known mechanical properties such as fused quartz.^[2]

The contact depth h_c is defined by the following equation.

Where h_max is the maximum indentation depth (figure 2), P_max is the maximum load and S is the stiffness. Epsilon is related to tip geometry. For example, for the Berkovich tip the ϵ=0.75.^[3] If the tip geometry would not be included, the contact depth would be equal to the plastic depth (h_p) which would result in inaccurate contact area according to the Oliver and Pharr method.^[3]

Hardness is defined by the maximum load and the contact area.^[3]

The effective modulus E_f is determined by the stiffness and contact area.^[3]

Where B is related to the tip geometry (1.034 for Berkovich). It is assumed that the modulus is independent of the indentation depth.^[3] This is often untrue with thin films and coatings since the substrate may affect mechanical properties of the sample. Thus, the indentation depth is typically set to be approximately 10% of the film thickness to minimize the effect of the substrate.^[2]

The elastic modulus of the sample can be determined from the effective modulus.^[2]

Where E₁ is the elastic modulus of the sample, v₁ is Poisson's ratio of the sample, and, E₂ and v₂ are the elastic modulus and the Poisson's ratio of the indenter tip, respectively. Thus, the sample's Poisson's ratio must be known. However, an approximate value is often used (e.g. 0.2 for ceramics and 0.4 for polymers) as the exact value has often little impact on the modulus.^[1]

2. Experimental details

In this section, the main experimental aspects of nanoindentation are shortly discussed.

2.1. Equipment 

A standard measurement configuration includes a sample stage, an indenter, a transducer and a displacement sensor.^[1] The load is applied by the transducer, and the displacement (piezoelectric) sensor monitors the displacement of the indenter. Consequently, the applied load is obtained as a function of the displacement (depth). The measurement can be done in different ways, such as, by adjusting the maximum load or the maximum depth. The most common setup is to determine a suitable maximum load for the measurement. 

A typical nanomechanical test instrument for nanoindentation includes also setup for other, complementary, nanomechanical measurements. For example, widely used nanoscratch test can be utilized in order to measure adhesion and scratch resistance. Furthermore, nanoindentation measurements can be done together with in-situ imaging which provides both qualitative (shape, pile-ups) and quantitative (wear volume, crack length) information about the indentation. In in-situ imaging, the sample surface is scanned in a raster manner with the indenter tip before and after the indentation. Thus, an image of the surface topography is formed before and after the measurement. The in-situ imaging also provides possibility to choose the spot of the indentation extremely accurately (for example, on a certain structure).^[4]

2.2. Sample preparation

Nanoindentation requires minimal sample preparation and can be conducted for any type of solid materials such as organic, inorganic, composite, crystalline and amorphous material.^[1] However, the sample surface should be rather flat as roughness may have a major impact on the depth measurement.^[2] Thus, polishing the surface may be required before nanoindentation. In addition, the measurement provides only local information of the material since all data is based on a single indentation at a specific location. However, the measurement can be done multiple times for each sample, and to different locations.^[1]

2.3. Effect of environment

The resolution limit for force is in the scale of piconewtons and the displacement (depth) in the scale of subnanometers.^[1] As nanoindentation deals with such small-scale units, environmental control is crucial. Temperature fluctuation of the surroundings results in thermal expansion and contraction of the sample and indenter, which results in thermal drift of depth measurement.^[2] Thus, good control of the temperature of the surroundings is required, and hence, measurements are conducted in a sealed chamber. Also, humidity should be well controlled, and be less than 50 %.^[2] The equipment should also be on stable ground as the ground vibrations cause drift on the measurement, too.^[2]

2.4. Reliability of the indentation

Repeatability of indentations is critical in order to verify the reliability of the measurements. Often, the repeatability is studied by performing indentation with different maximum loads as the results (e.g. elastic modulus) should not vary as a function of the maximum load (i.e. as a function of the indentation depth).^[2] If the repeatability is good, each of the loading curves will follow the same curve, and the unloading curves should have regular spacing.^[2] Furthermore, the indentation can be performed at different areas of the sample in order to define the homogeneity of the surface.

3. Case study

SiO₂ films are widely used in micro-electro-mechanical systems (MEMS) due to their excellent optical properties and chemical stability, for example.^[5]Zhao et al.^[5] studied the effect of film thickness on hardness of SiO₂ thin films in order to obtain more comprehensive understanding of the mechanical properties of the widely used SiO₂ films. They conducted nanoindentation measurements for 500 nm, 1000 nm and 2000 nm thick SiO₂ films. The measurements were done with the Berkovich indenter at room temperature. Figure 4 shows the load-depth curves of each film with different maximum loads. 

Figure 4.^[5] a)-c) Load-depth curves of the 500, 1000 and 2000 nm thick films, respectively. The relative penetration depth is the ratio between the indentation depth and film thickness. The maximum load was held for 10 s between loading and unloading. License: CC BY 4.0.

Reliability of the measurements was very good.^[5] Each substrate was measured with various maximum loads (figure 4), each following the same loading curve with regular spacing at the unloading part. Furthermore, SiO₂ films clearly had elastic-plastic behavior since part on the maximum indentation was recovered during unloading, but the recovery was not complete.

Hardness as a function of relative penetration depth of each substrate is depicted in figure 5. It can be clearly seen that the hardness of the 500 nm thick film is highest, and the 1000 nm thick film has slightly higher hardness than the 2000 nm thick film.

Figure 5.^[5] Hardness as function of relative penetration depth. License: CC BY 4.0.

The variation of hardness is quite high with low relative penetration depth due to different error sources such as resolution of indenter, roughness of the sample and noise of the surroundings.^[5] Furthermore, the hardness starts to increase after ~0.6 relative depth which is caused by the substrate. Thus, the intrinsic hardness values were calculated from 0.4-0.5 relative depth, which was the most stable range.^[5] The intrinsic hardness values were 11.9, 10.7 and 10.4 GPa of the 500, 1000 and 2000 nm thick films, respectively. 

Grain size of the thin films increased as a function of film thickness, which was detected by SEM imaging. Since the intrinsic hardness decreased as a function of film thickness, it also decreased as a function of grain size, which is in line with the Hall-Petch relationship (smaller grains lead to smaller pile-ups which results in greater hardness).^[5] Figure 6 summarizes the effect of film thickness and grain size on hardness.

Figure 6.^[5] Intrinsic hardness as a function of film thickness and grain size. License: CC BY 4.0.

In conclusion, the nanoindentation provided hardness values with very good reliability (figure 4). Furthermore, the hardness was dependent on the film thickness due to the effect of the grain size.^[5] 

4. References