Defect-induced multicomponent electron scattering in single-walled carbon nanotubes

We present a detailed comparison between theoretical predictions on electron scattering processes in metallic single-walled carbon nanotubes with defects and experimental data obtained by scanning tunneling spectroscopy of Ar${}^{+}$ irradiated nanotubes. To this purpose, we first develop a formalism for studying quantum transport properties of defected nanotubes in the presence of source and drain contacts and a scanning tunneling microscopy tip. The formalism is based on a field theoretical approach describing low-energy electrons. We account for the lack of translational invariance induced by defects within the so-called extended k$·$p approximation, which allows for multicomponent scattering with new scattering channels that are associated with exchanged momenta larger than the difference between the $K$ points of the nanotube. The theoretical model reproduces the features of the particle-in-a-box-like states observed experimentally. Further, the comparison between theoretical and experimental Fourier-transformed local density of states maps yields clear signatures for intervalley and intravalley electron scattering processes depending on the tube chirality.

Figure 1

(a) The real-space lattice of a carbon nanotube. The strip rolled into a cylinder is displayed as an orange area. The vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are a basis vector set for the honeycomb lattice, and ${\mathbf{C}}_h$ and $\mathbf{T}$ are the circumferential and the translational vector of the SWNT, respectively. (b) The reciprocal space of a carbon nanotube is represented by the red line segments. The figure correspond to a $(4,1)$ SWNT—a metallic tube. Here, ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$ are the reciprocal-lattice basis vectors for the honeycomb lattice. (c) Energy spectrum for the same SWNT. The thick red lines show the low-energy bands investigated here.

Figure 2

Reciprocal space for a (4,1) SWNT in the case of an impurity breaking the translational invariance. Here $k_A$ and $k_C$ are the axial and circumferential axes, respectively. The effective size ${\ell }_s$ of the scatterer in panel (a) is bigger than in panel (b).

Figure 3

Outline of the experimental setup and of the Fabry-Pérot resonator model.

Figure 4

Sketch of elastic scattering events by a local defect in a chiral (9,0) CNT within the extended $\mathbf{k}·\mathbf{p}$ approximation. The parallel (yellow) lines are due to the periodic boundary condition along the tube circumference; their length is determined by the spatial extend of the defect. The (red) circle is the Fermi surface of the valleys at the energy of the scattered electrons. Available right (left) moving electron states are showed as green (blue) dots. The defect scatters electrons between all states indicated as dots. Some typical processes are showed explicitly. The green arrow shows an intravalley backscattering event. The blue arrow depicts an intervalley backscattering process. Finally, the black arrow indicates an extended intervalley forward scattering event.

Figure 5

(a) A portion of a metallic SWNT about $40$ nm long exposed to a low dose of 1.5 keV Ar${}^{+}$ ions showing two defects labeled d1 and d2. (b) Detailed image of the zone delimited by d1 and d2. (c) $\mathit{dI}/\mathit{dV}$ scan recorded along the horizontal dashed line in (b). (d) $|\mathit{dI}/\mathit{dV}(k,V)|{}^2$ map resulting from a line-by-line zero padding FFT performed between the positions indicated by the two red arrows drawn in (c). (e) Calculated differential conductance for a (14,5) SWNT with interdefect length $L=10.4$ nm and asymmetric impurity strengths ${\lambda }_R=0.35$ and ${\lambda }_L=0.15$. (f) Corresponding $|\mathit{dI}/\mathit{dV}(k,V)|{}^2$ map.

Figure 6

STM topography image and corresponding $\mathit{dI}/\mathit{dV}$ scan of a metallic SWNT exposed to 200 eV Ar${}^{+}$ ions. (a) Situation with two defect sites. The upper panel shows a raw STM image of an about 30-nm-long metallic SWNT with three defect sites; the lower panel shows the same data after plane correction and flattening. (b) A situation in which two cuts have been performed at the defect positions by means of voltage pulses of $\approx \left|5\right|$ V. (c) A third voltage pulse has been performed close to the left end. The color scale of the $|\mathit{dI}/\mathit{dV}(k,V)|{}^2$ map is the same for the three cases. $V_s=0.5$ V, $I_s=0.3$ nA, $T=5.22$ K, $x_{\mathrm{res}}=0.15$ nm. (d) Calculated differential conductance with corresponding $|\mathit{dI}/\mathit{dV}(k,V)|{}^2$ map for a (10,4) SWNT with interdefect length $L=20$ nm and symmetric impurity strengths ${\lambda }_{R,L}=0.3$.