Complex band structure calculations based on the overbridging boundary matching method without using Green's functions

A complex band structure describes the dispersion relation not only of propagating bulk states but also of evanescent ones, both of which are together referred to as generalized Bloch states and are important for understanding the electronic nature of solid surfaces and interfaces. On the basis of the real-space finite-difference formalism within the framework of the density functional theory, we formulate the Kohn-Sham equation for generalized Bloch wave functions as a generalized eigenvalue problem without using any Green's function matrix. By exploiting the sparseness of the coefficient matrices and using the Sakurai-Sugiura projection method, we efficiently solve the derived eigenvalue problem for the propagating and slowly decaying/growing evanescent waves, which are essential for describing the physics of surface/interface states. The accuracy of the generalized Bloch states and the computational efficiency of the present method in solving the eigenvalue problem obtained are compared with those by other methods using the Green's function matrix. In addition, we propose two computational techniques to be combined with the Sakurai-Sugiura projection method and achieve further improvement in the accuracy and efficiency. Complex band structures are calculated with the present method for single- and multiwall carbon nanotubes, and the interwall hybridization and branch points of evanescent electronic states observed in the imaginary parts of the band structures are also discussed.

Figure 1

Schematic representation of crystalline solid with translational symmetry. The unit cells are indexed as $\cdots ,i-1,i,i+1,\cdots$. The matrix ${\mathbf{H}}^{\left(i\right)}$ is the Hamiltonian block corresponding to the $i\mathrm{th}$ unit cell, and the matrix $\mathbf{B}$ represent the interaction between the neighboring unit cells. The curves with arrows represent a propagating Bloch wave and decaying evanescent Bloch wave. The wave-function vector $\mathbf{\psi }$ is also partitioned by the unit cells.

Figure 2

Computational errors in solving eigenvalue problems for generalized Bloch states of carbon nanotubes. Computational errors are evaluated using the Euclidean norms of the residual vectors $\mathit{r}$, which are obtained by substituting the computed eigenpair into Eqs. (3) and (8). Eigenpairs are calculated using the SS-Hankel and SS-RR methods. The number of sampling points for the numerical contour integral and maximum order of moment used in the SS-RR method, $N_{\text{int}}$ and $N_{\text{mom}}$, are set to 32 and 16, respectively. The number of right-hand-side vectors, $N_{\text{rhs}}$, is set to 128 for (6,6) and 56 for (8,0) carbon nanotubes. The thick, normal, and thin dashed horizontal lines represent the averages of the computational errors over the all eigenpairs indicated by the crosses, solid circles, and open circles, respectively.

Figure 3

Average numbers of BiCG iterations until convergency. Equation (12) is solved for two different carbon nanotubes. Data points represented with solid symbols are plotted as a logarithmic function of domain radius $\left|z\right|$, and each data set is well fitted by a biquadratic function, as shown in dashed curve.

Figure 4

(a) Schematic representation of dividing an annular domain, and (b) [(c)] computational error in $\lambda$ and $\mathbf{\psi }$ for a (6,6) [(8,0)] carbon nanotube. The open circles in (a) represent sampling points on the integral paths $C_2$ and $C_3$, which are $z_{{\text{A}}^{\prime }}=1/z_{\text{A}}^{*}$. The subdomains are represented by the gray areas, and the dashed circle the unit circle on the complex plane. The computational errors are evaluated by the Euclidean norms of the residual vectors of Eq. (8), i.e., $\parallel (\lambda {\mathbf{\Pi }}_{\mathbf{B}}-{\mathbf{\Pi }}_{\mathbf{A}}){\mathbf{\psi }\parallel }_2$. The parameters for the Sakurai-Sugiura projection method are set as $N_{\text{int}}=32$ and $N_{\text{mom}}=16$. $N_{\text{rhs}}$ is 128 and 56 for (6,6) and (8,0) carbon nanotubes, respectively. In the case with the dividing-and-scaling technique, $N_{\text{rhs}}$ is reduced to 32 and 14 for (6,6) and (8,0) carbon nanotubes, respectively. The thick, normal, and thin dashed horizontal lines represent the averages of the computational errors over the all eigenpairs for the cases without the dividing-and-scaling technique, with the dividing-and-scaling technique, and with the dividing-and-scaling and interchange techniques, respectively.

Figure 5

Complex band structures of (a) (6,6) armchair carbon nanotube and (b) (8,0) zigzag carbon nanotube. Complex band structures composed of black dots are obtained with the present method, and band structures composed of open circles are obtained from conventional electronic structure calculations. $E_{\text{F}}$ denotes the Fermi energy.

Figure 6

Complex band structures of single-wall zigzag carbon nanotubes. Solid dots are obtained with the present method, and open circles from conventional electronic structure calculations. (a) and (b) are the complex band structures of (17,0) and (26,0) carbon nanotubes, respectively.

Figure 7

Complex band structures of multiwall carbon nanotube. Multiwall nanotube is composed of (8,0), (17,0), and (26,0) carbon nanotubes. (b) is an enlarged view around the origin of (a). Solid dots are obtained with the present method, and open circles from conventional electronic structure calculations. Notable data points are in red and are connected with lines for visibility.

Figure 8

Complex band structures of multiwall heteronanotube and two single-wall carbon nanotubes. The multiwall nanotube is composed of (8,0) and (17,0) carbon nanotubes and (26,0) boron-nitride nanotube. Solid dots are obtained with the present method, and open circles from conventional electronic structure calculations. Some data points are connected with solid and dotted curves for visibility. B and B′ in (a) indicate the branch points of the bands. The spatial distributions of the electronic states at A, B, and C are depicted in Fig. 9.

Figure 9

Spatial distributions of generalized Bloch states of multiwall hetero nanotube. In each panel, the wave function values on an $xy$ plane are plotted in logarithmic scale. Note that the wave functions depicted in the panel (b) are multiplied by a phase, $exp\left(i\theta \right)$, so as to become real numbers. (a), (b), and (c) correspond to A, B, and C indicated in Fig. 8, respectively. Since the generalized Bloch states in each of (a), (b), and (c) are doubly degenerate, the two wave functions are depicted in each panel. Concentric dashed circles represent (8,0) and (17,0) carbon, and (26,0) boron-nitride nanotube walls. Dashed lines are the guides to see the symmetry of the wave functions.

Figure 10

Line profiles of generalized Bloch states across the multiwall heteronanotube. In each panel, the amplitude of the wave function on the line $y=0$ in Fig. 9 is plotted in logarithmic scale as a function of the position $x$. The solid and dotted curves in (a), (b), and (c) represent the line profiles of the left and right panels in Figs. 9–9, respectively. The positions of the respective nanotube walls are indicated by the vertical dashed lines.