Valley coupling in finite-length metallic single-wall carbon nanotubes

Degeneracy of discrete energy levels of finite-length, metallic single-wall carbon nanotubes depends on the type of nanotubes, boundary condition, length of nanotubes, and spin-orbit interaction. Metal-1 nanotubes, in which two nonequivalent valleys in the Brillouin zone have different orbital angular momenta with respect to the tube axis, exhibit nearly fourfold degeneracy and small lift of the degeneracy by the spin-orbit interaction reflecting the decoupling of two valleys in the eigenfunctions. In metal-2 nanotubes, in which the two valleys have the same orbital angular momentum, vernier-scale-like spectra appear for boundaries of orthogonal-shaped edge or cap termination reflecting the strong valley coupling and the asymmetric velocities of the Dirac states. Lift of the fourfold degeneracy by parity splitting overcomes the spin-orbit interaction in shorter nanotubes with a so-called minimal boundary. Slowly decaying evanescent modes appear in the energy gap induced by the curvature of nanotube surface. Effective one-dimensional lattice model reveals the role of boundary on the valley coupling in the eigenfunctions.

Figure 1

Cutting lines for (a) $(n,m)=(6,3)$, and (b) $(7,4)$ SWNTs. In each figure, shadow areas show three different choices of the 2D BZ of graphene. $S_1{S}_2$ short segments denote conventional cutting lines, while $L_1{L}_2$ Long segments are another definition of 1D BZ. Here, $d=3,d_R=3,T=\sqrt{21}a,|{\mathit{C}}_h|=3\sqrt{7}a$, and $N=42$ for $(6,3)$ SWNT, and $d=1,d_R=3,T=\sqrt{31}a,|{\mathit{C}}_h|=\sqrt{93}a$, and $N=62$ for $(7,4)$ SWNT.

Figure 2

Boundary shape, calculated energy levels, and eigenstates for $(7,4)$ nanotube of $50.17$ nm length with the ends of orthogonal boundary. (a) Unfolded tube near the left end. The empty sites are represented by the dashed circles, and the carbon atoms at the boundary are marked by the solid circles. The red site (also indicated by $\mathrm{K})$ represents the Klein-type termination. The shadowed areas repeat the structure in the unshadowed area. (b) Energy levels ${\varepsilon }_l$ in $-35\le l\le 35$. $l$ is the energy level index numbered in ascending order of the energy and $l=0$ corresponds to the HOMO level. (c) Level separation ${\varepsilon }_{l+1}-{\varepsilon }_l$, as a function of $l$. The levels of $-1\le l\le 2$ indicated by red color in (b) and (c) are slowly decaying modes. (d)–(f) Local density (d) $l=-2$, (e) $l=-1$, and (f) $l=3$. Blue shows that for A sublattice, and red shows that for B sublattice. (g) Zone-folded intensity plot of Fourier transform of wave function on A sublattice for each level as a function of $k$. The energy for each level is added for each intensity plot. The intensities for states of spin-up majority are shown. They are presented by either solid or dashed lines in turn in increasing the energy to show them clearly. The blue lines show the energy band calculation under the periodic boundary condition. Right figure in (g) shows the energy levels of the even parity (blue lines) and the odd parity (red lines) for $V_{\mathrm{SO}}=0$. Note that the parity is not a good quantum number for the presence of the spin-orbit interaction, as will be discussed in Sec. 4d. The arrows with e (even) and o (odd) in (g) and in (b) show the states exhibiting intravalley coupling with the same parity. In (d)–(g), the components of orbital and spin are summed up for each site or each wave number.

Figure 3

Boundary shape, calculated energy levels, and eigenstates for $(7,4)$ nanotube of $50.17$ nm length with the ends of minimal boundary. (a) Unfolded tube near the left end. (b) Energy levels ${\varepsilon }_l$ in $-35\le l\le 35$. (c) Level separation ${\varepsilon }_{l+1}-{\varepsilon }_l$ as a function of $l$. The case of absence of spin-orbit interaction is shown by the red cross in the lower panel. (d)–(f) Local density for (d) $l=-4$, (e) $l=-3$, and (f) $l=5$. (g) Zone-folded intensity plot of Fourier transform of wave function on A sublattice for each level. The intensities for the states of spin-up majority are shown. (h) Enlarged plot of (g) near ${\varepsilon }_l=147$ meV (upper panel) and ${\varepsilon }_l=-142$ meV (lower panel). The case of absence of spin-orbit interaction is shown by the red dashed lines.

Figure 4

Length dependence of splitting of fourfold degeneracy for $(7,4)$ nanotube with minimal boundary. Three same symbols at each length show the averaged splitting in the energy range $|{\varepsilon }_l|<250$ meV except for the evanescent modes. The splittings are then also averaged for the three cases of the adjacent lengths $L_{\mathrm{NT}},L_{\mathrm{NT}}+a_z,L_{\mathrm{NT}}+2a_z$ because the splitting shows nearly threefold oscillations with respect to the nanotube length in the unit of $a_z=0.022$ nm. Solid line shows the averaged splitting with the spin-orbit interaction $V_{\mathrm{SO}}=6$ meV, dashed line shows that without the spin-orbit interaction. Inset shows the splitting as a function of the inverse of the length. The solid square with a bar (at $0.34$ meV) shows the absolute value of the spin splitting averaged in the energy range $|{\varepsilon }_l|<250$ meV calculated from the energy band calculation.

Figure 5

Effective one-dimensional model for $(n/d,m/d)=(7,4)$ SWNTs in which $(t_1,t_2)=(5,-6)$. The solid lines show the hopping between atoms. The red lines denote the hopping from (and to) A atom at site $\ell$.

Figure 6

Two examples of the left end classified into (a) orthogonal boundary condition in which $|t_2|$ A sites and $t_1$ B sites are empty, and (b) minimal boundary condition in which $n/d$ A sites and $m/d$ B sites are empty, for $(n/d,m/d)=(7,4)$ nanotubes. The dashed circles represent the empty atomic sites and the dashed lines represent the missing bonds. The solid circles represent the carbon atoms at the boundary, and the red circle represents the Klein-type termination.

Figure 7

Boundary shape, calculated energy levels, and eigenstates for $(6,3)$ nanotube of $50.15$ nm length with both ends minimal boundary as shown in (a). (b) Energy levels ${\varepsilon }_l$ in $-35\le l\le 35$. (c) Level separation ${\varepsilon }_{l+1}-{\varepsilon }_l$ as a function of $l$. The dashed lines in the lower panel show the spin-orbit splitting for corresponding energy calculated by the energy band calculation. (d)–(f) Local density for (c) $l=-4$, (d) $l=-3$, and (e) $l=5$.

Figure 8

Boundary shape and calculated energy levels ${\varepsilon }_l$ for $(6,3)$ nanotube of $50.63$ nm length. (a) Unfolded tube near the left (lower) and right (upper) ends. The upper end is cut along the $n{\mathit{a}}_1$ and $m{\mathit{a}}_2$. Note that both ends are classified into the minimal boundary. (b) Energy levels ${\varepsilon }_l$ in $-35\le l\le 35$. (c) Level separation ${\varepsilon }_{l+1}-{\varepsilon }_l$ as a function of $l$.

Figure 9

Boundary shape, calculated energy levels, and intensity plot in wave number for $(6,6)$ armchair nanotube of $50.05$ nm length. (a) Unfolded tube near the left end. (b) Energy levels ${\varepsilon }_l$ in $-35\le l\le 35$. (c) Level separation ${\varepsilon }_{l+1}-{\varepsilon }_l$ as a function of $l$. (d) Intensity plot of Fourier transform of wave function on A sublattice for each level. The intensities for the states of spin-up majority are shown. The blue lines show the energy band calculation under the periodic boundary condition.

Figure 10

Boundary shape, calculated energy levels, and intensity plot in wave number for $(7,4)$ nanotube of $49.56$ nm length with both ends capped. (a) 3D and (b) 2D representation of the cap structure. In (b), the solid circles indicate the A and B atoms connected to the cap region, and the open circles indicate the carbon atoms in the cap region at the left end. (c) Energy levels ${\varepsilon }_l$ in $-35\le l\le 35$. (d) Level separation ${\varepsilon }_{l+1}-{\varepsilon }_l$ as a function of $l$. (e) Intensity plot of Fourier transform of wave function on A sublattice for each level. Right figure in (e) shows the energy levels of the even parity (blue lines) and the odd parity (red lines) for $V_{\mathrm{SO}}=0$. The arrows in (c) and (e) show the states exhibiting intravalley coupling.

Figure 11

Coordinates for $(7,4)$ nanotube. The dashed lines show the interval of the 1D lattice constant $a_z=Td/N$. For this case, $d=1,T=\sqrt{31}a$, and $N=62$.