Topological edge and corner states in the biphenylene network

The electronic states and topological properties of the biphenylene network (BPN) are analyzed using a tight-binding model based on the $\pi$-electron network. It is shown that tuning the hopping parameters induces topological phase transitions, leading to the emergence of edge states owing to the nontrivial topological Zak phase of the bulk BPN. Elementary band analysis clearly gives the number of edge states, which are associated with the location of Wannier centers. In addition, we present the conditions for the emergence of corner states owing to the higher-order topological nature of the BPN.

Figure 1

(a) The lattice structure of the BPN. The shaded region is the unit cell, with six nonequivalent sublattices of carbon atoms (A to F). ${\mathit{a}}_1=(a,0)$ and ${\mathit{a}}_2=(0,a_T)$ are two primitive vectors, where $a$ and $a_T=\frac{3a}{\sqrt{3}+1}$. The intracellular (intercellular) hoppings denoted by red (black) lines are $\gamma$ (${\gamma }^{\prime }$). (b) The first BZ of the BPN and its high-symmetry points, $\mathrm{\Gamma }=(0,0), X=(\pi /a,0), Y=(0,\pi /a_T)$, and $S=(\pi /a,\pi /a_T)$.

Figure 2

(a) The energy band structure and the corresponding DOS for ${\gamma }^{\prime }/\gamma =1$. There are six subbands. (b) Three-dimensional plot of the energy band structure for bands 2, 3, and 4. (c) The energy band structure along the $k_x$ direction. Tilted Dirac dispersions can be seen on the $\mathrm{\Gamma }-\mathrm{X}$ line between bands 2 and 3. (d) The energy band structure along the $k_y$ direction. Band 3 is flat along the $\mathrm{\Gamma }-Y$ line, and band 4 has a parabolic dispersion at the $\mathrm{\Gamma }$ point.

Figure 3

Energy band structure of the 2D BPN for (a) ${\gamma }^{\prime }/\gamma =1/3$, (b) ${\gamma }^{\prime }/\gamma =1/2$, (c) ${\gamma }^{\prime }/\gamma =1.5$, and (d) ${\gamma }^{\prime }/\gamma =2$. The band degenerating points are marked with green circles.

Figure 4

(a) Schematic lattice structure of the BPN ribbon (BPR) with zigzag edges. $N$ is the width of the ribbon. The translational vector $\mathit{T}$ for the zigzag BPR is defined as $\mathit{T}=T(1,0)$. Here, $\mathit{T}=m{\mathit{a}}_1+n{\mathit{a}}_2=T(m,n)$. (b) The first BZ of the zigzag BPR (thick magenta line). The shaded yellow area is the first BZ of the 2D BPN. In a zigzag BPR, its first BZ is projected onto the $k_x$ axis and given as $-\pi /a\le k_x\le \pi /a$. The integration path $C$ for the calculation of the Zak phase is taken along $k_y$. (c) Energy band structure for the zigzag BPR for $N=50$. Cyan lines indicate the modes of TESs that emerge in the energy gaps where the total Zak phase is $\pi$. (d) The corresponding Zak phase of the zigzag BPR for the lowest three subbands, where the Zak phase is finite in the second and third subbands. (e) Schematic lattice structure of a BPR with armchair edges. $N$ is the width of the ribbon. The translational vector $\mathit{T}$ for the armchair BPR is defined as $\mathit{T}=T(0,1)$. (f) The first BZ of the zigzag BPR (thick magenta line). The shaded yellow area is the first BZ of the 2D BPN. In an armchair BPR, its first BZ is projected onto the $k_y$ axis and given as $-\pi /a_T\le k_y\le \pi /a_T$. The integration path $C$ for the calculation of the Zak phase is taken along $k_x$. (g) Energy band structure for the armchair BPR for $N=50$. Cyan lines indicate the modes of TESs that emerge in the energy gaps where the total Zak phase is $\pi$. (h) The corresponding Zak phase for the armchair BPR for the lowest three subbands, where the Zak phase is finite in the first and the third subbands.

Figure 5

Energy band structure of the BPR and the corresponding Zak phase. (a) Zigzag BPR with ${\gamma }^{\prime }/\gamma =1/2$ (left) and ${\gamma }^{\prime }/\gamma =2$ (right). In these cases, TESs (cyan lines) emerge in the energy gap where the Zak phase is $\pi$. (b) Armchair BPR with ${\gamma }^{\prime }/\gamma =1/2$ (left) and ${\gamma }^{\prime }/\gamma =2$ (right). TESs shown by cyan lines appear in the energy gap where the Zak phase is $\pi$. However, TESs shown by magenta lines emerge in the energy band gap where the Zak phase is $2\pi$. The puzzle will be resolved by considering Wannier centers (see the text).

Figure 6

The irreps of the occupied energy bands for ${\gamma }^{\prime }>\gamma$. Note that only the lowest three energy bands of the 2D BPN are displayed.

Figure 7

(a) Wyckoff positions (WPs) on the $xy$ plane for space group $Pmmm$. WPs $2i, 2k, 2m$, and $2o$ are two positions on the green lines. In the case of ${\gamma }^{\prime }$ (gray line) > $\gamma$ (red line), the three occupied bands of the BPN consist of two elementary bands, ${\mathit{b}}_{17}$ (WP $1e$) and ${\mathit{b}}_{45}$ (WP $2o$) in Table 2. (b) Top: Since the edge-terminated line of the zigzag BPR crosses the Wannier orbital (indicated by the blue ellipse) once, one edge state appears. Bottom: On the other hand, the edge-terminated line of the armchair BPR crosses the Wannier orbitals (the red ellipse) twice, which results in two edge states.

Figure 8

The structure of the BPN nanoflake for ${\gamma }^{\prime }>\gamma$. The red ellipses indicate the Wannier orbitals. Deleting the atom basis indicated by the black dots at the four corners, there are ($m\times n\times 12-6$) atom sites. Wannier orbitals are cut by the corner geometries, resulting in uncoupled Wannier orbitals at the corner sites.

Figure 9

(a) The energy spectrum of the BPN nanoflake. (b) The amplitude of the wave function of the six energy levels in the gap. Two states are localized in the left and right corners, and the other four states are localized in the upper and lower corners.

Figure 10

(a) Energy band structures obtained with DFT (dashed lines) and wannier90 (solid lines). In the wannier90 calculation, only $p_z$ orbitals of carbon atoms are taken into account for the projection. (b) Schematic of the BPN lattice structure with the definition of nearest-neighbor (NN) hoppings for the tight-binding model of the 2D BPN. The yellow shaded rectangle is the unit cell. ${\gamma }_0$ is the intracellular NN hoppings for the bonds of B-C and E-F. ${\gamma }_0$ is the intracellular NN hoppings for the bonds of A-B, C-D, D-E, and F-A. ${\gamma }_x^{\prime }$ and ${\gamma }_y^{\prime }$ are the intercellular hoppings for the $x$ and $y$ directions, respectively. (c) The definition of second- and third-NN electron hoppings. Magenta thick dashed lines indicate the second-NN hoppings. ${\tau }_0$ is the intracellular hoppings for the bonds of A-E and A-C. ${\tau }_y^{\prime }$ is the intercellular hoppings for the bonds of A-E and A-C. ${\tau }_x^{\prime }$ is the intercellular hoppings for the bonds of A-B and A-F. Cyan dotted lines indicate the third-NN hoppings. ${\zeta }_0$ is the intracellular hoppings for the bond of A-D. ${\zeta }_1$ is the intracellular hoppings for the bonds of B-E and C-F. ${\zeta }_y^{\prime }$ is the intercellular hoppings for the bonds of B-C and E-F. Actual values of the hoppings are summarized in Table 3. (d) Energy band structure of the effective tight-binding model considering up to the NN hopping (dashed lines) and the third-NN hopping (solid lines). Inclusion of up to third-NN hoppings reproduces the energy band structures obtained with DFT shown in (a) sufficiently well.