A class of topological nodal rings and its realization in carbon networks

Topological nodal rings can be classified into three types according to the slopes in their energy dispersion. Type-I nodal rings consist of type-I points, type-II rings consist of type-II points, and type-III rings consist of anisotropic points. Taking carbon networks as an example, we show that all three types can exist and transform from one to another under experimentally realizable strain. Within a given type, a transition between a topological metal phase and a semiconductor phase also takes place, and each type of nodal rings shrinks into the corresponding type of semi-Dirac points, including novel type-II and -III semi-Dirac points. These topological features are expected to exhibit diverse electron-hole pocket patterns and Landau levels, giving rise to unusual transport properties.

Figure 1

Atomic structures of (a) CNW-(3,2) and (b) CNW-(1,2), in which the common $n=2$ pink nanoribbons run along $z$. They belong to the same series of CNW-(m,n), formed by connecting the armchair ribbons in (c) of width $m$ and $n$. Both blue ${\mathrm{C}}_1$ and pink ${\mathrm{C}}_2$ balls are $s{p}^2$ atoms, while gray balls are $s{p}^3$ atoms. (d) Conventional (top) and primitive (bottom) cells of CNW-(1,2). Symbols $t_1$ and $t_2$ in (c) and $t_3,t_4$, and $t_5$ in (d) denote hopping parameters.

Figure 2

Three types of SDPs and nodal rings. (a)–(c) Type-I, -II, and -III SDPs. The red (solid), blue (dashed), and olive (dotted) lines denote the energy bands along $k_x,k_y$, and $k_z$, respectively. (d)–(f) Type-I, -II, and -III topological nodal rings at $k_z=0$. They are shown as green (dotted) circles, which intersect the (equienergy) Fermi level at the four yellow points.

Figure 3

Band structure projected onto atomic orbitals: pink corresponds to ${\mathrm{C}}_2 p_{\perp }$ orbitals and blue corresponds to ${\mathrm{C}}_1 p_{\parallel /\perp }$ orbitals. (a),(b) CNW-(3,2): (a) under no strain. Inset is the first BZ, where the green circle denotes the location of the type-I nodal ring. (b) Under a compressive +2.3% strain along $z$. Inset is the charge density plot for states near the type-I SDP. (c),(d) CNW-(1,2): (c) under no strain. Inset is a band structure near the type-III SDP. (d) Under a compressive +2.2% strain along $z$. Inset is a contour plot of the type-III ring at $k_z=0$.

Figure 4

(a)–(c) EHPs for type-I, -II, and -III nodal rings. Light blue and orange denote electron ($E$) and hole ($H$) pockets, respectively. Green dotted circles and yellow points have the same meanings as in Figs. 2–2. (d)–(f) LLs for type-I, -II ($\gamma =0.5$), and -III ($\delta =0.5$) nodal rings as a function of magnetic field $B$. In the calculation, we set $k_z=0$ and $\mathrm{\Delta }=0.7$ eV.