We studied the lateral contact stiffness (kcontactx) between the tip of a frictional force microscope and a pillar surface to identify the sliding behavior of the tip at the submicrometer scale. The kcontactx and mean lateral force (⟨F⟩) were systematically measured as functions of pillar diameter φ. We found that the kcontactx on a single Si pillar surface increased whereas the ⟨F⟩ rapidly decreased as φ decreased from the micrometer to the nanometer scale. This contradiction could be explained by the change in tip behavior from dynamic sliding to static sticking.

Frictional force microscopy (FFM) is an important tool for studying atomic-scale friction, adhesion, lubrication, and wear.1 However, this technique is not always able to provide correct information on frictional behaviors when the mechanical stiffness of the sample material changes,2 because of corresponding changes in the contact mechanics between the instrument tip and sample. For example, when the static friction between the tip and sample is small, only the dynamic friction can be measured in the scan region. However, when the static friction is large, static friction or the stick–slip phenomenon may occur in the scan region. Thus, changes in material properties require investigation of the frictional behavior using as many measurement methods as possible.

Miura et al.3,4 recently reported that size effects, such as softening, occur in MoS2 when its island size decreases from the micrometer to the submicrometer scale. Strong elastic softening in silicon nanotubes has also been observed by other groups.5–7 In this paper, FFM is used to measure the lateral force and contact stiffness on a single Si pillar surface as a complementary technique to understand the change in contact mechanics between the tip and pillar surface as a function of the Si pillar size.

In our study, an FFM system was used to assess the frictional force on the surface of a single pillar ranging in size from the micrometer to the nanometer scale. Lateral forces F and force loops were measured at a loading force N of 1 nN under a relative humidity of <50% at room temperature using a commercially available FFM instrument (SPI-3700; Seiko Instruments, Inc.) at a scan velocity v of 64 nm/s. A rectangular silicon cantilever with a Si3N4 tip (normal spring constant: 0.75 N/m) was used for the F and force loop measurements. The measured F was directly calibrated using crossed dual cantilevers.8 

Figure 1(a) shows a schematic of pillar arrays with pillar diameters φ of 70 nm, 200 nm, 1 μm, and 3 μm fabricated on a silicon wafer. Each pillar had a height h of ∼200 nm, and each pillar array had an area of 3 × 3 mm2. Figure 1(b) shows a scanning electron microscopy (SEM) image of a nanopillar array with φ of 100 nm and interpillar spacing d of 5 μm.

FIG. 1.

Schematic of cylindrical pillars and scanning electron microscopy (SEM) images of pillar arrays. (a) Schematic of cylindrical pillar arrays with pillar diameters φ of (A) 70 nm, (B) 200 nm, (C) 1 µm, and (D) 3 µm fabricated on a (E) silicon wafer. The pillar height h is ∼200 nm. (b) SEM image of pillar arrays with φ of 100 nm and interpillar spacing d of 5 µm.

FIG. 1.

Schematic of cylindrical pillars and scanning electron microscopy (SEM) images of pillar arrays. (a) Schematic of cylindrical pillar arrays with pillar diameters φ of (A) 70 nm, (B) 200 nm, (C) 1 µm, and (D) 3 µm fabricated on a (E) silicon wafer. The pillar height h is ∼200 nm. (b) SEM image of pillar arrays with φ of 100 nm and interpillar spacing d of 5 µm.

Close modal

First, the nanoscale sliding behavior of single pillars with diameters of 70 nm φ 3 μm was studied. Figures 2 and 3 show the dependence of φ on the mean F, ⟨F⟩, and the effective lateral stiffness of the cantilever–tip–pillar system, keffx. keffx is estimated by the slope of the lateral force loop. The F, when the sliding length x is 2 nm, as shown in the insets in Fig. 2, was measured at N = 1 nN and v = 64 nm/s.

FIG. 2.

Mean lateral force ⟨F⟩ observed on each single pillar surface as a function of the pillar diameter φ. Changes in ⟨F⟩ and the lateral force loop (insets) on each single pillar surface as a function of φ were observed by FFM. ⟨F⟩ was evaluated from the Fx plots of the lateral force loops; black and red lines indicate forward and backward tip scans, respectively. The scan velocity was v = 64 nm/s, and the loading force was N = 1 nN.

FIG. 2.

Mean lateral force ⟨F⟩ observed on each single pillar surface as a function of the pillar diameter φ. Changes in ⟨F⟩ and the lateral force loop (insets) on each single pillar surface as a function of φ were observed by FFM. ⟨F⟩ was evaluated from the Fx plots of the lateral force loops; black and red lines indicate forward and backward tip scans, respectively. The scan velocity was v = 64 nm/s, and the loading force was N = 1 nN.

Close modal
FIG. 3.

Effective lateral stiffness keffx observed on single pillar surfaces as a function of the pillar diameter φ. The keffx of the cantilever–tip–pillar system was evaluated from the slope of the Fx plots of the lateral force loops shown in the insets in Fig. 2. The scan velocity was v = 64 nm/s, and the loading force was N = 1 nN. The elastic penetration depth δ increased as φ decreased.

FIG. 3.

Effective lateral stiffness keffx observed on single pillar surfaces as a function of the pillar diameter φ. The keffx of the cantilever–tip–pillar system was evaluated from the slope of the Fx plots of the lateral force loops shown in the insets in Fig. 2. The scan velocity was v = 64 nm/s, and the loading force was N = 1 nN. The elastic penetration depth δ increased as φ decreased.

Close modal

We then evaluated ⟨F⟩ by averaging the lateral force loops shown in the insets of Fig. 2. As φ decreased from 3 μm to 70 nm, ⟨F⟩ rapidly decreased from 1 nN to 10 pN. As φ decreased further to the submicrometer range, ⟨F⟩ approached zero but keffx increased. When the scan length of the lateral force loop was increased from 2 to 10 nm in the FFM experiment, no change in keffx was observed when 70 nm φ 3 μm.

In contrast to ⟨F⟩ in Fig. 2, keffx in Fig. 3 rapidly increased with decreasing φ for the sliding length x = 2 nm. Specifically, keffx was 0.65, 1.0, 2.2, 2.6, 3.3, and 3.3 N/m when φ was 3 µm, 1 µm, 500 nm, 300 nm, 100 nm, and 70 nm, respectively.

The effective lateral stiffness keffx of the cantilever–tip–pillar system is defined as9 

(1)

where kcontactx, ktipx, kpx (Table I),10 and kleverx are the lateral stiffness values of the tip–pillar contact region, tip, pillar, and cantilever, respectively. Using kleverx = 650 N/m,9 ktipx = 39 N/m,9 and kcontactx=(1/keffx1/ktipx1/kpx1/kleverx)1, we estimated kcontactx to be 0.662 N/m when φ = 3 μm and 3.79 N/m when φ = 70 nm. These kcontactx values are very close to the keffx obtained when φ = 3 μm (0.65 N/m) and φ = 70 nm (3.3 N/m), as shown in Fig. 3, since kcontactx is negligibly small compared to ktipx, kpx, and kleverx for all φ considered. Thus, the increase in keffx may be ascribed to the increase in kcontactx.

TABLE I.

Spring constant kpx values of pillars as a function of the pillar diameter φ. The lateral spring constant kpx of the cylindrical pillar with φ, h = 200 nm, and the Young’s modulus of silicon, E = 185 GPa, was calculated using the relation.10 

φ 70 nm 100 nm 300 nm 500 nm 1 µ3 µ
kpx 81.8 N/m 341 N/m 27.6×kN/m 213 kN/m 3.41 MN/m 276 MN/m 
φ 70 nm 100 nm 300 nm 500 nm 1 µ3 µ
kpx 81.8 N/m 341 N/m 27.6×kN/m 213 kN/m 3.41 MN/m 276 MN/m 

Because the frictional force loop shows hysteresis between the forward and backward scans when φ = 3 μm, the FFM tip may be expected to slide on the pillar surface within the tip scan region. Therefore, ⟨ F⟩ ≅1 nN when φ = 3 μm represents the “dynamic” frictional behavior. However, because the frictional force loop shows no hysteresis between the forward and backward scans when φ = 70 nm, the FFM tip may be expected to stick to the pillar surface within the tip scan region. Therefore, ⟨F⟩ ≅10 pN and kcontactx=3.79 N/m when φ = 70 nm represents the “static” frictional behavior. These observations indicate that the tip behavior changes from dynamic sliding to static sticking when the pillar diameter decreases from the micrometer to the nanometer scale.

In this study, kcontactxkeffx can be defined as the force per unit displacement required to shear an elastic contact region in a particular direction. For sphere–plane contact, kcontactx is given by11,12

(2)

where a is the contact radius and G*=(2ν1)/G1+(2ν2)/G21. In G*, G1, and G2 are the shear moduli of the tip and sample, respectively, and ν1 and ν2 are the corresponding Poisson’ ratios. In the case of Hertzian contact, a is given by a=3RN/4E*1/3, where R, N, and E* are the tip sphere radius loading force, and effective Young’s modulus, respectively. Here, E* is given by E*=(1ν12)/E1+(1ν22)/E21, where E1 and E2 are the Young’s moduli of the tip sphere and sample plane, respectively.

Brazil and Pharr13 reported that if the stick–slip transition does not occur during sliding, kcontactx is proportional to the elastic penetration depth δ or a for various materials, which indicates that G* is constant. Thus, the increase in kcontactx with decreasing φ can be ascribed to the increase in a when G* is constant. A previous report on ultrathin crystalline silicon nanotubes5–7 revealed that Young’s modulus, E, prominently decreased as the diameter of the nanotubes decreased, which indicated that a soft area expanded on the silicon nanotubes. Similarly, in the present study, as the φ/h aspect ratio of the pillar decreased, an elastic soft area grew on the Si pillar surface. This area expansion led to an increase in a between the tip and Si soft area because a1/E*1/3. In this case, the FFM tip penetrated deeply into the Si soft area on the pillar surface, and a large δ and kcontactx were induced. Thus, kcontactx increased as φ decreased, as shown in Fig. 3.

In this study, nanoscale friction on a single pillar was investigated using FFM. The marked effects of φ on kcontactx and ⟨F⟩ were observed for pillar diameters of 70 nm φ 3 μm. When φ = 3 μm, ⟨F⟩ represents the dynamic frictional force due to FFM tip sliding. However, when φ = 70 nm, ⟨F⟩ and kcontactxkeffx represent the static frictional force because the FFM tip sticks to the pillar surface. Thus, as the pillar size decreased to the nanometer scale, the tip behavior changed from dynamic sliding to static sticking. Future studies on macroscopic systems with multiple pillar arrays will benefit from an improved understanding of the changes in frictional behavior due to changes in contact mechanics from a single pillar to multiple pillars.

We gratefully acknowledge the financial support provided by the Eco-Project of Aichi University of Education and Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant Nos. 17K05054, 17H02785, 21H01747, and 26390064. We would like to thank Editage (www.editage.com) for English language editing.

The authors have no conflicts to disclose.

Y.I. collected all of the experimental data. R.Y., N.W., and M.N. prepared the pillar samples. M.N. and N.S. collaborated in discussions with K.M. K.M. conceptualized the research and wrote the paper and N.S. supported writing it.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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