Simulating atomic force microscope images with density functional theory: The role of nonclassical contributions to the force

We discuss a scheme for calculating atomic force microscope images within the framework of density functional theory (DFT). As in earlier works [T. L. Chan et al., Phys. Rev. Lett. 102, 176101 (2009); M. Kim and J. R. Chelikowsky, Appl. Surf. Sci. 303, 163 (2014)] we do not simulate the cantilever explicitly, but consider it as a polarizable object. We go beyond previous studies by discussing the role of exchange and correlation effects; i.e., we approximately take into account the Pauli interaction between sample and cantilever. The good agreement that we find when comparing our calculated images to experimental images for the difficult case of the 8-hydroxyquinoline molecule demonstrates that exchange-correlation effects can play an important role in the DFT-based interpretation of AFM images.

Figure 1

Left: Orientation of the sample (8-hydroxyquinoline) in the $x,y$ plane. Right: Oscillating cantilever in the $z$ direction, with $z_0$ as the rest position, $A$ as the amplitude of the oscillation, and $\mathrm{\Delta }z$ as the distance between the surface of the sample and the cantilever.

Figure 2

Upper figure: Geometry of the 8-hydroxyquinoline molecule, with carbon in black, hydrogen in gray, nitrogen in blue, and oxygen in red. Lower figure: Experimental AFM images from Ref. [9]. Reprinted with permission from AAAS. From the left to the right the rest position of the cantilever was decreased.

Figure 3

Calculated AFM image of 8-hydroxyquinoline using $v_{\text{TS}}$ according to Eq. (6). The images show the shift of the frequency of the cantilever $\mathrm{\Delta }f$ as a function of the position $(x,y)$ of the cantilever for a rest position of $z_0$ = 4.8 $a_0$ (top) and $z_0$ = 7 $a_0$ (bottom). A general trend emerges: When the rest position of the cantilever is increased, the spotlike structure disappears. The parameter $A$ was chosen as $A$ = 2 $a_0$ in analogy to the experimental data in Ref. [9].

Figure 4

Calculated AFM image of 8-hydroxyquinoline using $v_{\text{TS}}$ according to Eq. (7). The images show the shift of the frequency of the cantilever $\mathrm{\Delta }f$ as a function of the position $(x,y)$ of the cantilever for a rest position of $z_0$ = 4.8 $a_0$ (top) and $z_0$ = 7 $a_0$ (bottom). The parameter $A$ was chosen as $A$ = 2 $a_0$ in analogy to the experimental data of Ref. [9].

Figure 5

Calculated AFM image of 8-hydroxyquinoline with $v_{\text{TS}}$ according to Eq. (7) and using EXX-KLI for the exchange-correlation potential. The image shows the shift of the frequency of the cantilever $\mathrm{\Delta }f$ as a function of the position $(x,y)$ of the cantilever for a rest position of $z_0$ = 6 $a_0$ and an amplitude of $A$ = 2 $a_0$.

Figure 6

Density of states of 8-hydroxyquinoline. Red solid line: EXX. Blue dash-dotted line: LDA. For ease of comparison the eigenvalues were shifted such that ${\varepsilon }_{\text{HOMO}}$ = 0 eV in both cases.

Figure 7

Density of states of pentacene. Red solid line: EXX. Blue dash-dotted line: LDA. The eigenvalues are shifted such that ${\varepsilon }_{\text{HOMO}}$ = 0 eV.

Figure 8

Calculated AFM image of pentacene using $v_{\text{TS}}$ according to Eq. (7) and the LDA. The image shows the shift of the frequency of the cantilever $\mathrm{\Delta }f$ as a function of the position $(x,y)$ of the cantilever for a rest position of $z_0$ = 6 $a_0$. The parameter $A$ was chosen as $A$ = 2.1 $a_0$ in analogy to the 8-hydroxyquinoline calculations.

Figure 9

Calculated AFM image of pentacene with $v_{\text{TS}}$ according to Eq. (7) and using EXX-KLI for the exchange-correlation potential. The image shows the shift of the frequency of the cantilever $\mathrm{\Delta }f$ as a function of the position $(x,y)$ of the cantilever for a rest position of $z_0$ = 6 $a_0$ and an amplitude of $A$ = 2.1 $a_0$.

Figure 10

The image shows the density distribution of pentacene as a function of $(x,y)$ at $z=6 a_0$ above the $x,y$ plane.