Revealing Subsurface Vibrational Modes by Atom-Resolved Damping Force Spectroscopy

We propose to use the damping signal of an oscillating cantilever in dynamic atomic force microscopy as a noninvasive tool to study the vibrational structure of the substrate. We present atomically resolved maps of damping in carbon nanotube peapods, capable of identifying the location and packing of enclosed $\mathrm{Dy}@{\mathrm{C}}_{82}$ molecules as well as local excitations of vibrational modes inside nanotubes of different diameter. We elucidate the physical origin of damping in a microscopic model and provide quantitative interpretation of the observations by calculating the vibrational spectrum and damping of $\mathrm{Dy}@{\mathrm{C}}_{82}$ inside nanotubes with different diameters using ab initio total energy and molecular dynamics calculations.

Figure 1

(a) Schematic of an oscillating AFM tip scanning across nanotube peapods with different diameters $d$ at different filling levels. (b) Total charge density of the optimized $\mathrm{Dy}@{\mathrm{C}}_{82}$ metallofullerene encapsulated in the $(18,0)$ nanotube, superposed with the skeletal structure of the peapod. (c) Schematic of the coupling mechanism between a sharp AFM tip and a flexible substrate. The cantilever stiffness is represented by the spring connecting the tip and its suspension. The elastic substrate is represented by the spring connecting a representative atom to its surroundings. (d) Mechanism of energy dissipation during the dynamic AFM operation in noncontact regime.

Figure 2

AFM topography and damping in SWNTs and peapods of different diameter $d$. (a) Atomically resolved dynamic AFM observations of the topography (1–4) and damping (5–8), defined as energy loss per cycle $\Delta E$ and represented by the color-coded contour plots. Results of simultaneous scans for an empty SWNT with $d=16.2\pm 0.5\AA$ are shown in (1,5), those for peapods with $d=16.2\pm 0.3\AA$ in (2,6), with $d=15.0\pm 0.5\AA$ in (3,7), and with $d=13.0\pm 0.5\AA$ in (4,8). (b) Topographic and damping profiles across the same nanotube peapod. The topographic data $z(x)$, represented by the black line, are taken along the black-white dotted trace in (a) (4). The damping data $\Delta E(x)$, represented by the red circles, are taken along the red-white dotted trace in (a) (8). (c) Relationship between nanotube diameter $d$ and the maximum damping energy $\Delta E_{\max }$. The data points are representative for tens of tube samples. The dashed line is a guide to the eye.

Figure 3

Calculated dynamics and energy dissipation in the ${\mathrm{C}}_{82}@\mathrm{SWNT}$ system following an initial “radial plucking” at the nanotube surface. Strongest damping occurs when plucking an atom located axially $\approx 1\AA$ off the center of the nearest ${\mathrm{C}}_{82}$. (a) Radial displacement of the plucked atom as a function of time in ${\mathrm{C}}_{82}@(18,0)$. (b) Radial displacement of the ${\mathrm{C}}_{82}$ center of mass as a function of time in ${\mathrm{C}}_{82}@(18,0)$. (c) Fourier spectrum of the radial displacement of the initially plucked atom as a function of the nanotube diameter $d$. (d) Fourier spectrum of the ${\mathrm{C}}_{82}$ center of mass as a function of the nanotube diameter $d$. The insets in (a) and (b) illustrate the initial distortion and the quantity monitored.