Dirac Cones, Topological Edge States, and Nontrivial Flat Bands in Two-Dimensional Semiconductors with a Honeycomb Nanogeometry

We study theoretically two-dimensional single-crystalline sheets of semiconductors that form a honeycomb lattice with a period below 10 nm. These systems could combine the usual semiconductor properties with Dirac bands. Using atomistic tight-binding calculations, we show that both the atomic lattice and the overall geometry influence the band structure, revealing materials with unusual electronic properties. In rocksalt Pb chalcogenides, the expected Dirac-type features are clouded by a complex band structure. However, in the case of zinc-blende Cd-chalcogenide semiconductors, the honeycomb nanogeometry leads to rich band structures, including, in the conduction band, Dirac cones at two distinct energies and nontrivial flat bands and, in the valence band, topological edge states. These edge states are present in several electronic gaps opened in the valence band by the spin-orbit coupling and the quantum confinement in the honeycomb geometry. The lowest Dirac conduction band has $S$-orbital character and is equivalent to the $\pi \text{−}{\pi }^{\star }$ band of graphene but with renormalized couplings. The conduction bands higher in energy have no counterpart in graphene; they combine a Dirac cone and flat bands because of their $P$-orbital character. We show that the width of the Dirac bands varies between tens and hundreds of meV. These systems emerge as remarkable platforms for studying complex electronic phases starting from conventional semiconductors. Recent advancements in colloidal chemistry indicate that these materials can be synthesized from semiconductor nanocrystals.

Popular Summary

Fueled by our unquenchable thirst for electronic devices that are ever smaller in size and more versatile in functions, materials physicists’ search for new classes of two-dimensional (2D) materials with remarkable electronic properties has not stopped at the discoveries of single atomic-layer graphene and 2D topological insulators. Engineering—with known materials—artificial materials with similar or even richer and more exotic electronic properties appears to be a natural avenue to explore. In this theoretical paper, we bring some excitement to this exploration and show that, remarkably, an artificial 2D material—a honeycomb lattice formed by nanocrystals of very conventional semiconductors—can be either graphenelike or topological-insulator-like electronically.

Based on atomistic calculations of electronic structure of a honeycomb lattice of nanocrystals of zinc-blende semiconductors, we show that this artificial material, whose lattice structure is the same as that of graphene, has an amazingly rich electronic structure, hosting several fundamentally different electronic properties. One part of the electronic structure shows the bulk-insulating and edge-conducting characteristics of a topological insulator, while another part exhibits two Dirac cones, similar to those in graphene, as well as nontrivial “flat bands.” These flat bands could be of great importance to the realization of the elusive fractional quantum spin Hall effect.

The assembly of the kind of 2D artificial nanocrystal lattices we have investigated has already become experimentally possible recently. Our study, therefore, heralds the emergence of a wide variety of materials engineered to have variable nanogeometry and rich, tunable electronic structures and functions.

Figure 1

(a),(b) Block models for the self-assembled honeycomb lattices based on nanocrystals that have a truncated cubic shape. (a) Honeycomb lattice formed by atomic attachment of three of the 12 $\{110\}$ facets (light green and orange regions) corresponding to a graphene-type honeycomb structure (both sublattices in one plane). (b) Honeycomb lattice formed by attachment of three of the eight $\{111\}$ facets (dark green and red regions), leading to the silicene-type configuration (each sublattice in a different plane). In both models, a $⟨111⟩$ direction is perpendicular to the honeycomb plane, as is experimentally observed. The arrow indicates the superlattice parameter $d$. (c) Top view of the unit cell of a graphene-type honeycomb lattice of PbSe nanocrystals. (The Pb atoms are represented by gray spheres, the Se atoms by yellow spheres.).

Figure 2

(a) Lowest conduction bands and (b) highest valence bands of a graphene-type honeycomb lattice of truncated nanocubes of CdSe (body diagonal of 4.30 nm). The zero of energy corresponds to the top of the valence band of bulk CdSe. (c) Energy splittings (meV) between the two highest valence bands (VBs), between the two $1S$ Dirac conduction bands ($1S$), and between the two $1P$ Dirac conduction bands ($1P$) calculated at $\mathbf{K}+\mathbf{q}$ and plotted using polar coordinates, i.e., energy versus the angle between $\mathbf{K}$ and $\mathbf{q}$. (The squares and red curve correspond to $|\mathbf{q}|=0.05|\mathbf{K}|$, the circles and blue curve to $|\mathbf{q}|=0.15|\mathbf{K}|$, and the triangles and black curve to $|\mathbf{q}|=0.25|\mathbf{K}|$.)

Figure 3

Evolution of the bandwidth of the $1S$ Dirac band as a function of the number of atoms that form the nanocrystal-nanocrystal contact ($N_{\text{at}}$). The size of the nanocrystals, which also determines the period of the honeycomb lattice, is indicated by the body diagonal of the truncated nanocube. Results are shown for the graphene geometry (bonding via the $\{110\}$ facets).

Figure 4

Highest valence bands in ribbons made from a graphene-type honeycomb lattice of truncated nanocubes of CdSe (body diagonal of 4.30 nm). (a) Ribbon with zigzag edges (ribbon $\text{width}=57\mathrm{nm}$, periodic cell length $l=8.2\mathrm{nm}$). (b) Ribbon with armchair edges (ribbon $\text{width}=33\mathrm{nm}$, periodic cell length $l=14.2\mathrm{nm}$). In each case, the unit cell is composed of 16 nanocrystals (34768 atoms per unit cell). The colored regions indicate the bands of the corresponding 2D semiconductor, i.e., those shown in Fig. 2.

Figure 5

2D plots of the wave functions of the edge states calculated at $k=0.3\times 2\pi /l$ for the ribbon considered in Fig. 4 (energy of the states equaling approximately $-0.123\mathrm{eV}$). The plots are restricted to a single unit cell of the ribbon [(a) higher-energy state, (b) lower-energy state]. The vertical axis corresponds to the direction of the ribbon. More than 90% of each wave function is localized on the nanocrystals at the edges. The white dots indicate the atoms.

Figure 6

Evolution of the energy gap between the two highest valence bands of CdSe sheets with graphenelike honeycomb geometry as a function of the number of atoms that form the nanocrystal-nanocrystal contact ($N_{\text{at}}$). The size of the nanocrystals is indicated by the body diagonal of the truncated nanocube.

Figure 7

Chirality of the $1S$ wave functions of the CdSe honeycomb lattice compared with atomic chirality in graphene. (a) Phase shift (radians) on each atomic orbital between the electronic states of the lowest conduction band of the graphene-type honeycomb superlattice of the compound CdSe [Fig. 2]. The phase shift is calculated at $\mathbf{K}+{\mathbf{q}}_1$ and $\mathbf{K}+{\mathbf{q}}_2$, with $|{\mathbf{q}}_1|=|{\mathbf{q}}_2|=0.05|\mathbf{K}|$ and an angle between ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ of 3.8 rad, for each lateral atomic position in the lattice plane. (b) Difference between the phase of the wave function at the center of nanocrystals $A$ and $B$ versus the angle between ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$, at $K$ (square blue symbols) and $K^{\prime }$ (circular red symbols). Similar results are obtained for the $1P$ Dirac cone.

Figure 8

(a) Lowest conduction bands of a silicene-type honeycomb lattice of truncated nanocubes of CdSe (body diagonal of 5.27 nm). (b) Evolution of the gap at $K$ between the $1S$ (blue squares) and between the $1P$ (red circles) Dirac bands versus the electric field strength applied perpendicular to the honeycomb plane.

Figure 9

(a) Lowest conduction bands and (b) highest valence bands of a graphene-type honeycomb lattice of truncated nanocubes of PbSe (body diagonal of 5.00 nm).

Figure 10

Band structure calculated for a silicene-type lattice using the effective tight-binding Hamiltonian for three sets of parameters. (a) $E_s(A,B)=2.18\mathrm{eV}$, and all the $E_p=2.44\mathrm{eV}$, $V_{ss\sigma }=-8\mathrm{meV}$, $V_{sp\sigma }=0\mathrm{meV}$, $V_{pp\sigma }=50\mathrm{meV}$, and $V_{pp\pi }=-2\mathrm{meV}$. (b) Same as (a) but with $E_{p_z}(A,B)=2.51\mathrm{eV}$. (c) $E_s(A)=2.18\mathrm{eV}$, $E_s(B)=2.19\mathrm{eV}$, $E_{p_x}(A)=E_{p_y}(A)=2.44\mathrm{eV}$, $E_{p_x}(B)=E_{p_y}(B)=2.46\mathrm{eV}$, $E_{p_z}(A)=2.51\mathrm{eV}$, $E_{p_z}(B)=2.52\mathrm{eV}$, $V_{ss\sigma }=-8\mathrm{meV}$, $V_{sp\sigma }=0\mathrm{meV}$, $V_{pp\sigma }=50\mathrm{meV}$, and $V_{pp\pi }=-2\mathrm{meV}$.

Figure 11

Lowest conduction bands of a graphene-type honeycomb lattice of spherical nanocrystals of CdSe ($\text{diameter}=4.7\mathrm{nm}$). Nearest-neighbor nanocrystals are connected by cylindrical bridges of CdSe (diameter of the $\text{cylinders}=2.3\mathrm{nm}$).

Figure 12

Highest valence bands of a graphene-type honeycomb lattice of truncated nanocubes of CdSe (body diagonal of 4.30 nm) under a magnetic field $\mathbf{B}$ that induces Zeeman splitting of the bands (${\mu }_B{g}_{S}B=1\mathrm{meV}$). The corresponding bands at zero magnetic field are those shown in Fig. 2. The Chern numbers of the bands are indicated in the figure.