Aharonov-Bohm interference and beating in deformed single-walled carbon nanotube interferometers

The quantum conductance modulations induced by an axial magnetic field for deformed single-walled carbon nanotube (SWNT) interferometers under two types of deformation i.e., uniaxial tensile and torsional strains, have been studied in the tight-binding approximation. We have analytically derived the expression for the beating modulation period, which is caused by the Aharonov-Bohm interference between two nondegenerate subbands of spiraling electrons. It is found that the beating pseudoperiod is very sensitive to the type and strength of the applied strains, and also the chiral angle of the SWNT. For example, under tensile strain, all metallic SWNTs have the beating modulation except the armchair ones, while under torsional strain, only metallic zigzag SWNTs have no beating modulation. Therefore, the beating modulation could be a promising powerful tool to precisely measure experimentally the degee of nanoscale mechanical deformation of the deformed carbon nanotubes.

Figure 1

(Color online) A two-dimensional model of an armchair nanotube electron resonator. The atoms on a highlighted ring construct an interface.

Figure 2

Structure of a single graphene sheet. ${\vec{r}}_1$, ${\vec{r}}_2$, and ${\vec{r}}_3$ correspond to bonds 1, 2, and 3, respectively. ${\vec{a}}_1$ and ${\vec{a}}_2$ are the lattice vectors of the two-dimensional sheet. $\widehat{c}$ and $\widehat{t}$ are unit vectors along the circumference and nanotube axis, respectively.

Figure 3

(Color online) Left panel: plot of conductance $G$ vs $V_g$ for the elongated (15,15) armchair nanotube with $\alpha =0.01$, $u_1=1.0\mathrm{eV}$, $u_2=6.0\mathrm{eV}$, $M=6496$ (corresponding to a nanotube length of about $800\mathrm{nm}$), and ${\epsilon }_t=0.01$, in different axial magnetic fields. Solid (black) dotted (red), and dashed (blue) lines correspond to $B=0$, 2, and $4\mathrm{T}$, respectively. (Note: the shift of conductance along $V_g$ with increasing $B$ is small and cannot be distinguished.). Right panel: 2D plot of conductance $G$ vs $V_g$ and $B$ for the same nanotube; the magnetic field ranges from $-45\text{to}45\mathrm{T}$.

Figure 4

(Color online) The same plots as Fig. 3, but for elongated zigzag nanotube (27,0) with a length of $800\mathrm{nm}$ (the dashed lines in right panel reflect the changing direction of the conductance peaks).

Figure 5

(Color online) Left panel: The first Brillouin zone of a metallic SWNT (in zero magnetic field). The black lines representing the two $k_{\Vert }$ lines of a perfect metallic SWNT cross the two Fermi points $K_1$ and $K_2$, while the dotted (red) lines represent the two $k_{\Vert }$ lines of the deformed metallic SWNT. The deformation causes the zero-band gap states to deviate from $K_1$ and $K_2$ by $\pm \Delta k_{\perp }$, opening a band gap. Right panel: the same as the left panel except in an axial magnetic field. Here, $\Delta k_{\perp }^1=\Delta k_{\perp }+2\pi \Phi ∕\mid \vec{R}\mid {\Phi }_0$, and $\Delta k_{\perp }^2=\Delta k_{\perp }-2\pi \Phi ∕\mid \vec{R}\mid {\Phi }_0$, and so the degeneracy is lifted.

Figure 6

(Color online) The same plots as Fig. 4, but for twisted zigzag nanotube and $\gamma =1°$.

Figure 7

(Color online) The same plots as Fig. 3, but for twisted armchair nanotube and $\gamma =1°$.