Suppression of the pseudoantisymmetry channel in the conductance of telescoped double-wall nanotubes

The conductance of telescoped double-wall nanotubes (TDWNTs) composed of two armchair nanotubes $\left[\right(n_O,n_O)$ and $(n_O-5,n_O-5)$ with $n_O⩾10$] is calculated using the Landauer formula and a tight binding model. The results are in good agreement with the conductance calculated analytically by replacing each single-wall nanotube with a ladder, as expressed by $(2e^2∕h)(T_{+}+T_{-})$, where $T_{+}\left(T_{-}\right)$ is the transmission rate of the pseudosymmetry (pseudoantisymmetry) channel. Interlayer hopping of the $-$ channel in the ladder model explains two key results: the low value of $T_{-}$ for TDWNTs except when $n_O=10$, and the particularly low value of $T_{-}$ when either $n_O$ or $n_O-5$ is a multiple of three. In the latter case, $T_{-}$ becomes zero when the saturation of the number of interlayer bonds per atom is neglected.

Figure 1

Telescoped double-wall nanotube.

Figure 2

Index $(l,i)$ representing $z$ and $\theta$ in a (2,2) armchair nanotube.

Figure 3

Schema of a telescoped nanotube and amplitude of the wave functions ${\vec{\psi }}_l^{\left(O\right)}$ and ${\vec{\psi }}_l^{\left(I\right)}(\Delta z<0)$.

Figure 4

Saturation effect of the interlayer hopping integral. When the distance between $i$ in tube $O$ and $j$ in tube $I$ is smaller than $d_0^2+r_0^2$, the atoms are connected by an AA bond. This threshold is represented by dashed ovals. The other interlayer bonds of an atom with an AA bond are weakened due to saturation of the number of covalent bonds per atom. The hopping integral between $i$ and $j$ in (c) and (d) is smaller than that in (b) for the same sites $i$ and $j$.

Figure 5

Schema of ladder model and amplitude of the wave functions ${\vec{\varphi }}_l^{\left(O\right)}$ and ${\vec{\varphi }}_m^{\left(I\right)}$. (a) The hopping integrals between the two ladders $h(O,l,i\mid I,m,j)$ are indicated by dashed lines and dotted lines for $i\ne j$ and $i=j$, respectively. For simplicity, the dotted lines are shown only for $l=-1$ and $m=-1,-2$. (b) Mirror planes corresponding to $\sigma =+,-$ and $\tau =u,d$. (c) Ladder model after unitary transformation ${\tilde{\varphi }}_{l,\sigma }^{\left(\mu \right)}=(1∕\sqrt{2})({\varphi }_{l,0}^{\left(\mu \right)}+\sigma {\varphi }_{l,1}^{\left(\mu \right)})$, where $\sigma =+,-$ and $\mu =O,I$.

Figure 6

(Color online) Conductance of the $(n_O,n_I)i-j$ TDWNT at $E=0.1\mathrm{eV}$.

Figure 7

(Color online) Conductance of the (10,5) TDWNT at $E=0.1\mathrm{eV}$ calculated using the tube model for $\Delta \theta =-\pi ∕500,-3\pi ∕1000,0,\pi ∕500,\pi ∕250$, and $\Delta z=0$. The transmission rate of the pseudoantisymmetry channel is recovered as $\mid \Delta \theta \mid$ increases.

Figure 8

(Color online) Conductance of the (10,5) $2\text{−}(-2)$ TDWNT at $E=0.1\mathrm{eV}$ calculated using the tube model for various parameters $W_{\mathrm{BB}}$ and $W_{\mathrm{AB}}$ determining the strength of the interlayer transfer integral in Eq. (6).

Figure 9

(Color online) Conductance of the (10,5) TDWNT at $E=0.1\mathrm{eV}$ calculated using the tube model for the exceptional interlayer configurations $(\Delta \theta ,\Delta z)=(0,0),(\pi ∕30,0)$. When $L_c$ in Eq. (6) is small, the interlayer transfer integral is short range, and the transmission rate of the pseudoantisymmetry channel is large even for these configurations.

Figure 10

(Color online) Conductance of (a) the (10, 5) 0-0 TDWNT and (b) the (10, 5) $2\text{−}(-2)$ TDWNT as a function of energy (calculated using the tube model).

Figure 11

(Color online) Honeycomb lattice of the (15,10) TDWNT near $z=0$, with $\theta$ as the vertical axis and $z$ as the horizontal axis, for (a) $0⩽\theta ⩽2\pi ∕3$, (b) $2\pi ∕3⩽\theta ⩽4\pi ∕3$, and (c) $4\pi ∕3⩽\theta ⩽2\pi$. $\Delta z$ is positive and $\Delta \theta =-\pi ∕15$. Gray curved lines represent interlayer bonds between $(O,1,6p-1)$ and $(I,2,4p)$, and black curved lines represent those between $(O,1,6p-11)$ and $(I,2,4p-7)$, with integers $p=1$, 2,…5. These bonds are labeled with corresponding indices of $\theta$ ($6p-1,4p,6p-11$, and $4p-7$). The curved lines with identical vertical position cancel in the interchain hopping integral of the pseudoantisymmetry channel. Gray and black interlayer bonds denote AB and BB bonds, respectively.

Figure 12

Vertical position of $(O,1,i)$ sites and $(I,2,j)$ sites in Fig. 11 with respect to $i$ and $j$.

Figure 13

(Color online) Conductance of the tube model at $E=0.1\mathrm{eV}$ calculated using the tube model for various values of $W_{\mathrm{BB}}-W_{\mathrm{AB}}$ while $W_{\mathrm{BB}}$ is fixed. The saturation effect becomes more pronounced as $W_{\mathrm{BB}}-W_{\mathrm{AB}}$ increases.