Degeneracy breaking and intervalley scattering due to short-ranged impurities in finite single-wall carbon nanotubes

We present a theoretical study of degeneracy breaking due to short-ranged impurities in finite, single-wall, metallic carbon nanotubes. The effective mass model is used to describe the slowly varying spatial envelope wave functions of spinless electrons near the Fermi level at two inequivalent valleys ($K$-points) in terms of the four component Dirac equation for massless fermions, with the role of spin assumed by pseudospin due to the relative amplitude of the wave function on the sublattice atoms (“$A$” and “$B$”). Using boundary conditions at the ends of the tube that neither break valley degeneracy nor mix pseudospin eigenvectors, we use degenerate perturbation theory to show that the presence of impurities has two effects. First, the position of the impurity with respect to the spatial variation of the envelope standing waves results in a sinusoidal oscillation of energy level shift as a function of energy. Second, the position of the impurity within the hexagonal graphite unit cell produces a particular $4\times 4$ matrix structure of the corresponding effective Hamiltonian. The symmetry of this Hamiltonian with respect to pseudospin flip is related to degeneracy breaking and, for an armchair tube, the symmetry with respect to mirror reflection in the nanotube axis is related to pseudospin mixing.

Figure 1

The positions with respect to the graphite unit cell of the perturbing potential are labelled as $A$, $B$, $C$, and $D$. Carbon atomic positions are at the six corners of the hexagon, there are three $A$ atomic positions $\{A_1,A_2,A_3\}$ and three $B$ atomic positions $\{B_1,B_2,B_3\}$. We also consider the potential to be near the center of the unit cell $\left(C\right)$ or at one of six positions half-way between neighboring atoms ($D_1$ to $D_6$). An additional small deviation $\delta \mathbf{R}$ of the potential position is shown (greatly exaggerated) for the $C$ position, with direction described by angle $\chi$ in the nanotube coordinates $(x,y)$. The figure has chiral angle $\eta =\pi ∕6$ corresponding to an armchair tube.

Figure 2

Plot of the modulus squared wave function ${\mid {\Psi }_1\mid }^2$ of the lowest states for diagonal boundary conditions: the solid lines show the second component ${\mid {\psi }_{BK}\mid }^2$ and the dashed lines show the first component ${\mid {\psi }_{AK}\mid }^2$. The lowest states $p_1=0,1,2,3$ are shown from top to bottom. Values of the mixing angles are taken to be ${\zeta }_m={\Lambda }_m=0$ so that the boundary conditions are satisfied by ${\psi }_{BK}=0$ at the ends of the nanotube $y=\pm L∕2$.

Figure 3

Relative amplitude of the wave function ${\Psi }_{\mathrm{tot},1}\left(\mathbf{r}\right)\propto \cos (\mathbf{K}\bullet \mathbf{r}\mp \pi ∕6)$ determined on the atomic sites, following Fig. 1(d) in Ref. 10. Dashed lines, labelled scan $A$ and scan $B$, are parallel to the tube axis.

Figure 4

Plot of the modulus squared total wave function ${\mid {\Psi }_{\mathrm{tot}}\mid }^2$ (arbitrary units) for off-diagonal boundary conditions that break valley degeneracy. The wave function is evaluated along line $A$ parallel to the axis of an armchair nanotube $\eta =\pi ∕6$, length $L=50a$. The four lowest energy states above the Fermi level are shown from top to bottom. Parameter values are $s=1$, ${\zeta }_m=\pi ∕4$, ${{\rm Y}}_p=-\pi ∕2$, and ${{\rm Y}}_m={\zeta }_p=0$.

Figure 5

Plot of the modulus squared total wave function ${\mid {\Psi }_{\mathrm{tot}}\mid }^2$ (arbitrary units) for off-diagonal boundary conditions that break valley degeneracy. The wave function is evaluated along line $B$ parallel to the axis of an armchair nanotube $\eta =\pi ∕6$, length $L=50a$. The four lowest energy states above the Fermi level are shown from top to bottom. Parameter values are $s=1$, ${\zeta }_m=\pi ∕4$, ${{\rm Y}}_p=-\pi ∕2$, and ${{\rm Y}}_m={\zeta }_p=0$.

Figure 6

Splitting $\delta E^{″}-\delta E^{\prime }$ of the pairs of degenerate energy levels of a clean nanotube Eq. (31) due to the effective Hamiltonian $\delta H_A$ of a perturbative potential on an $A$ atomic site. The symbols show the energy shift as a function of the energy of the unperturbed levels, solid lines are a guide for the eye. The upper curve is for the potential at $Y_0=0.025L$ (potential is one-twentieth of the way from the center of the nanotube to the end), lower curve is for $Y_0=0.125L$ (potential is a quarter of the way from the center of the nanotube to the end). $U_A=v_a^2{\phi }^2\left(0\right)U$ and parameter values are $s=1$, $\kappa =2\pi ∕3$, $\eta =\pi ∕6$, and ${{\rm Y}}_p={{\rm Y}}_m=0$.

Figure 7

Plot of the modulus squared total wave functions ${\mid {\Psi }_{\mathrm{tot}}\mid }^2$ (arbitrary units) at the Fermi level $(q=0)$ in the presence of an impurity on an atomic site, evaluated using degenerate perturbation theory. The wave functions are evaluated along line $B$ parallel to the axis of an armchair nanotube $\eta =\pi ∕6$, length $L=50a$. The standing wave corresponding to $\delta E=0$ is shown on top, that corresponding to $\delta E=V_{11}+V_{22}$ is below. Parameter values are $s=1$, $p=0$, and ${{\rm Y}}_p={{\rm Y}}_m={\zeta }_p={\zeta }_m=0$.

Figure 8

Splitting $\delta E^{″}-\delta E^{\prime }$ of the pairs of degenerate energy levels of a clean nanotube Eq. (35) due to the effective Hamiltonian $\delta H_A^{\prime }$ of a perturbative potential with a first order deviation $\delta \mathbf{R}$ from an $A$ atomic site. The symbols show the splitting as a function of the energy of the unperturbed levels, lines are a guide for the eye. The upper curve is for the potential at $Y_0=0.025L$ (potential is one-twentieth of the way from the center of the nanotube to the end), lower curve is for $Y_0=0.125L$ (potential is a quarter of the way from the center of the nanotube to the end). Parameter values are $s=1$, $\kappa =2\pi ∕3$, $\eta =\pi ∕6$, ${{\rm Y}}_p={{\rm Y}}_m=0$, and the angle of deviation of the potential is $\chi =\pi ∕4$.

Figure 9

Splitting $\delta E^{″}-\delta E^{\prime }$ of the pairs of degenerate energy levels of a clean nanotube Eq. (37) due to the effective Hamiltonian $\delta H_C$ of a perturbative potential near the center of the graphite unit cell. The symbols show the energy shift as a function of the energy of the unperturbed levels, solid lines are a guide for the eye. The upper curve is for the potential at $Y_0=0.025L$ (potential is one-twentieth of the way from the center of the nanotube to the end), lower curve is for $Y_0=0.125L$ (potential is a quarter of the way from the center of the nanotube to the end). Parameter values are $s=1$, $\kappa =2\pi ∕3$, $\eta =\pi ∕6$, ${{\rm Y}}_p={{\rm Y}}_m=0$, and the angle of deviation of the potential is $\chi =\pi ∕4$.

Figure 10

Position of energy levels as a function of the strength of a potential on an $A$ atomic site in the presence of degeneracy breaking due to off-diagonal boundary conditions ${\xi }_m=\pi ∕8$. The position of the impurity is $Y_0=0.125L$ (potential is a quarter of the way from the center of the nanotube to the end), and other parameter values are $\kappa =2\pi ∕3$, $\eta =\pi ∕6$, ${\xi }_p={{\rm Y}}_p={{\rm Y}}_m=0$.