Repulsive Vertical Forces

(a) (b)

0.0 0.5 1.0 1.5 2.0

380 390 400 410 420

Height (pm)

Line cut (nm)

Figure 4.18.: Deformation due to repulsive vertical forces. (a) 3D representations of constant total force landscapes (3.4 x 3.4 nm²) at -450 pN, -575 pN and -640 pN (top to bottom).

The color code ranges from 460 to 520, 370 to 450 and 355 to 410 pm for the three images, respectively. An indentation of the layer when high forces are applied is clearly visible in the rim region. (b) The intersections (black, red, green) of the light blue plane with the force landscapes shown in (a) underline the vertical compression, which can be estimated to be≈25 pm of the rim site by decreasing F– from -575 to -640 pN. Note that the height scale corresponds to the -640 pN curve and that the -575 and -450 pN curves are offset by -30 and -100 pm, respectively.

(a) (b)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 -100

-80 -60 -40 -20 -0

Vertical force (pN)

z (nm)

-100 -90 -80 -70 -60 -50 -40 0

10 20 30

Distance change (pm)

Vertical force (pN)

Figure 4.19.: (a) The short-range forces at the rim site are fitted by a Morse type force up to z=0.42 nm (gray dotted line), after which a significant change of slope is visible that is attributed to the pushing of the layer. (b) The difference between the fit and the data in (a) is used to deduce a vertical stiffness ofk_–=1.5 N/m.

Besides the lateral layer deformation the presence of the tip also impacts the shape of

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Figure 4.20.: (a) The short-range forces are fitted by a Lennard-Jones type force up toz=0.42 nm, after which the discrepancy of the fit and the data is attributed to the pushing of the layer. (b) By a linear fit a vertical stiffness ofk_–≈1.0 N/m is deduced from the data shown in (a).

Figure 4.21.: (a) A Morse type force is fitted to the data up to different cut-off-points (z = 0.40,0.42,0.47 nm, pink, black and green curve, respectively). A fit to the full range results in worse agreement (orange line). (b) The difference between the fit and the data in (c) is used to deduce vertical stiffnesses ofk_–=1.5,1.5 and 2.1 N/m, respectively.

the corrugation normal to the surface. Constant force landscapes interpolated from the vertical force component F– clearly show an indentation of the rim region in an area of about six hexagonal units of up to 25 pm at short tip-sample distances (Figure 4.18). This apparent indentation cannot be solely attributed to the topographic response of the layer to the acting force, as it also includes long-range electrostatic and van der Waals forces.

To get a quantitative measure of the deformation, focus is put on the distance dependence of the short-range forces. They are obtained by using the valley region as a reference for long-range forces and subtracting them from the total forces at the rim regions. Note that in the following only single F(z)-curves are discussed, which are obtained by averaging over several pixels in the region of interest from each slice of the set. (The spectroscopy curves used for the drift compensation in Section 4.3.1 can also be used, leading to the same results.)

4.3. Determining the Stiffness of h-BN

The fit of a Morse type force [173]:

F = −F0⋅ (e^−a(z−z0)−e^{−2a(z−z0)}) (4.3)

to the attractive part of the short-range forces results in F0 = 370 pN, a=12 nm⁻¹, and z₀=0.37 nm. This fit is interpolated and the discrepancy of it and the data be-low z=0.42 nm⁶ is attributed to topographic deformation of the layer. By relating the topographic changes to the acting vertical force an effective stiffness of k–=1.5±0.6 N/m is derived.

So far the fitting of the repulsive vertical forces is discussed in terms of a Morse type force.

Now an evaluation based on fitting a Lennard-Jones type force F–∝ (s¹²

z¹³) − (s⁶

z⁷) (4.4)

is presented, as shown in Figure 4.20 (a). In Figure 4.20 (b) the discrepancy of the fit and the data below z=0.42 nm is attributed to the softness of the layer and a stiffness of k–≈1.0 N/m can be derived, which fits well into the range of values obtained.

While so far the cut between the fit and the interpolation was set toz=0.42 (corresponding to the fourth height recorded in the data set) in Figure 4.21 (a) the effect of choosing different ranges for the Morse type force fit are shown. While the fit to the full range clearly shows the highest discrepancy for larger heights, the effect of choosing different cut-off points (e.g. the cyan curve for a cut-off at z=0.40 nm) has on a small effect on the stiffness (4.21 (b)). Even the relatively bad fit starting at z =0.47 nm (green curve) results in a comparable stiffness (within the error range) of 2.1 N/m.

Comparison to Indentation Measurements

To put the effective stiffness into perspective as far as larger scale measurements that usually aim at the Young’s modulus are concerned, it is worth while to compare the data presented here to the measurements of ref. [123]. They derive the 2D elastic modulus (Y^2D =200−500 GPa⋅nm) from indentation of 1µm diameter free few-layerh-BN mem-branes with a diamond tip. In their Figure 4c two indentation measurements are shown that allow the estimation of the indentation to be approximately 20−30 nm for an applied force of 50 nN. This would correspond to an effective stiffness of approximately 2 N/m.

Hence, the stiffness that is derived here for the rim regions ofh-BN on Rh(111) is of similar magnitude as for a much larger, free membrane.

In another recent publication aiming at the quantification of the stiffness of a graphene superstructure Koch et al. investigated graphene on Ru(0001) that forms ∼0.1 nm high

“nanodomes” with ∼3 nm periodicity [174]. Due to the observed indentation they ten-tatively estimate a stiffness of k_exp =0.65 N/m by attributing the slope of the repulsive short-range forces solely to the deformation, i.e. by neglecting the intrinsic change of force. In the above discussed model this problem is avoided by the comparison against

6Note that the position of the cut is arbitrary but can be justified by the fact that only the force range where repulsive forces become relevant is of interest. The small number of data points prohibits an analytical treatment as it is done in Section 4.3.5 for the attractive vertical forces.

the Morse type force. From extensive DFT calculations where the deformation is achieved by fixing one single atom at a reduced distance to the surface and relaxing the structure, they derive a comparably high stiffness of kDF T =43.6 N/m. Also this definition of the force is different from the approach discussed here, as they relate the deformation to the force acting on that single atom due to the restoring forces of the dome, while here the distance change is attributed to the short-range forces applied by the tip.