Theoretical study of solid iron nanocrystal movement inside a carbon nanotube

We use a first-principles based kinetic Monte Carlo simulation to study the movement of a solid iron nanocrystal inside a carbon nanotube driven by the electrical current. The origin of the iron nanocrystal movement is the electromigration force. Even though the iron nanocrystal appears to be moving as a whole, we find that the core atoms of the nanocrystal are completely stationary, and only the surface atoms are moving. Movement in the contact region with the carbon nanotube is driven by electromigration forces, and the movement on the remaining surfaces is driven by diffusion. Results of our calculations also provide a simple model, which can predict the center of mass speed of the iron nanocrystal over a wide range of parameters. We find both qualitative and quantitative agreement of the iron nanocrystal center of mass speed with experimental data.

Figure 1

Computed iron nanocrystal center of mass speed as a function of electromigration force $F_{\mathrm{em}}$. The simulation temperature is 700 K, somewhat larger than in the experiment,[1] the iron nanocrystal radius is $r_{\mathrm{cyl}}=1.05$ nm, and the length is $l=4.31$ nm. The line is a fit to a functional form as in Eq. (4).

Figure 2

Cross-section of an iron nanocrystal inside a carbon nanotube. Four regions of the nanocrystal are indicated (A, B, C, and D), see main text for details. For definiteness, the direction of the electromigration force is assumed to be to the right in this figure.

Figure 3

Two-dimensional projection of three-dimensional iron atom positions at eight different snapshots in the kinetic Monte Carlo simulation. Various intensities of greyness correspond to rows containing more (darker gray) or less (brighter gray) iron atoms. The simulation temperature is 700 K, somewhat larger than in the experiment,[1] the iron nanocrystal radius is $r_{\mathrm{cyl}}=1.11$ nm, and the length is $l=4.31$ nm. The electromigration force magnitude is $F_{\mathrm{em}}=0.28$ eV/nm.

Figure 4

Results from the same kinetic Monte Carlo simulation as in Fig. 3 in the form of a histogram indicating the number of iron atoms within each bin with length of one lattice constant ($a=0.29$ nm). The number of atoms initially in region A (B) are colored red (blue) and their positions are tracked during the simulation. Vertical position of the bars in the histograms are arbitrary, only the height of each individual bar (gray, red, or blue) is to be interpreted as the number of atoms within that bin.

Figure 5

Flow of iron atoms in the iron nanocrystal during its movement through the nanotube. See main text for a more detailed description. Larger flow magnitude is indicated with darker shade of gray, on a logarithmic scale. Direction of flow is shown with arrows (regardless of magnitude, all arrows have the same length). For clarity, all flow information is duplicated from positive radial coordinates to the negative coordinates. The simulation temperature in this calculation is 600 K, somewhat larger than in the experiment,[1] the iron nanocrystal radius is $r_{\mathrm{cyl}}=1.11$ nm, and the length is $l=4.31$ nm. The electromigration force magnitude is $F_{\mathrm{em}}=0.28$ eV/nm.

Figure 6

Dependence of iron nanocrystal center of mass speed on electromigration force (black) and total electrical current $I$ (red). Kinetic Monte Carlo simulation (black circles) was done for temperatures from 500 to 900 K in steps of 50 K, the iron nanocrystal radius for all calculations is $r_{\mathrm{cyl}}=1.05$ nm, while the length is $l=4.31$ nm. A fit to model from Eq. (4) with parameters given in Eq. (5) is shown with black lines for temperatures from 350 to 900 K in steps of 50 K. Experimental data from Ref. 1 are shown with red squares, and a fit to model from Eq. (4) to experimental data is shown with a red dashed line (the temperature used in the fit is 350 K, consistent with an independent experimental estimate). The relationship between electromigration force (bottom axis, black) and estimated current density $j$ through the iron nanocrystal (topmost axis, blue) is given by Eq. (6) as discussed in the text.

Figure 7

Two-dimensional projection of three-dimensional iron atom positions at eight different snapshots in the kinetic Monte Carlo simulation, as in Fig. 3. Various intensities of greyness correspond to rows containing more (darker gray) or less (brighter gray) iron atoms. The simulation temperature is 700 K, somewhat larger than in the experiment,[9] the iron nanocrystal radius in the region on the left is $r_{\mathrm{cyl}}=1.35$ nm, and on the right is $r_{\mathrm{cyl}}=1.11$ nm (as in Fig. 3). Therefore the cross-section area on the left is about 50$\%$ larger than on the right. The length of iron crystal at $t=0$ ms is $l=4.31$ nm. The electromigration force magnitude is $F_{\mathrm{em}}=0.28$ eV/nm.