# Dirac cones, topological edge states and non-trivial flat bands in two-
dimensional semiconductors with a honeycomb nano-geometry

E. Kalesaki  IEMN - Dept. ISEN, UMR CNRS 8520, Lille, France  Physics and
Materials Science Research Unit, University of Luxembourg, 162a avenue de la
Faïencerie, L-1511 Luxembourg  C. Delerue<EMAIL_ADDRESS>IEMN -
Dept. ISEN, UMR CNRS 8520, Lille, France  C. Morais Smith  Institute for
Theoretical Physics, University of Utrecht  W. Beugeling  Max-Planck-Institut
für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany  G.
Allan  IEMN - Dept. ISEN, UMR CNRS 8520, Lille, France  D. Vanmaekelbergh
Debye Institute for Nanomaterials Science, University of Utrecht

###### Abstract

We study theoretically two-dimensional single-crystalline sheets of
semiconductors forming 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 rock-salt 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 nano-geometry leads to rich band structures including, in the
conduction band, Dirac cones at two distinct energies and non-trivial 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 $\pi-\pi^{\star}$ 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.

##  I Introduction

The interest in two-dimensional (2D) systems with a honeycomb lattice and
related Dirac-type electronic bands has exceeded the prototype graphene Castro
Neto  _et al._ (2009). Currently, 2D atomic Lay  _et al._ (2009); Cahangirov
_et al._ (2010); Houssa  _et al._ (2011); Vogt  _et al._ (2012); Gomes  _et
al._ (2012) and nanoscale Park  _et al._ (2008); Park and Louie (2009); Simoni
_et al._ (2010); Gibertini  _et al._ (2009); Nádvorník  _et al._ (2012)
systems are extensively investigated in the search for materials with novel
electronic properties that can be tailored by geometry. For example, a
confining potential energy array with honeycomb geometry was created on a
Cu(111) surface and it was demonstrated that the electrons of the Cu surface
state have properties similar to those of graphene Gomes  _et al._ (2012).
From the same perspective, it was proposed that a honeycomb pattern with a
50-100 nm periodicity could be imposed on a 2D electron gas at the surface of
a conventional semiconductor by using lithography or arrays of metallic gates
Park  _et al._ (2008); Park and Louie (2009); Simoni  _et al._ (2010);
Gibertini  _et al._ (2009); Nádvorník  _et al._ (2012). Within the effective-
mass approach, linear $E({\bf k})$ relationships were predicted close to the
Dirac points in the Brillouin zone, in analogy with graphene Park  _et al._
(2008). The group velocity of the carriers was found to be inversely
proportional to the honeycomb period and to the effective carrier mass. In
order to obtain a system with sufficiently broad Dirac bands, it is thus of
major importance to reduce the period of the honeycomb lattices far below 50
nm and to use semiconductors with low effective mass.

Our aim in the present work is to explore theoretically the physics of 2D
semiconductors with honeycomb geometry and period below 10 nm. Electronic
structure calculations using the atomistic tight-binding method Jancu  _et
al._ (1998); Delerue and Lannoo (2004) attest that both the atomic lattice and
the overall geometry influence the band structure. We show that the honeycomb
nano-geometry not only enables the realization of artificial graphene with
tunable properties but also reveals systems with non-trivial electronic
structure which has no counterpart in real graphene.

We consider atomically-coherent honeycomb superlattices of rock-salt (PbSe)
and zinc-blende (CdSe) semiconductors. These artificial systems combine Dirac-
type electronic bands with the beneficial tunability of semiconductors under
strong quantum confinement. In the case of a zinc-blende atomic lattice,
separated conduction $1S$ and $1P$ Dirac cones of considerable bandwidth (10’s
to 100’s of meV) are found, as well as dispersionless $1P$ bands. Here, $1S$
and $1P$ refer to the symmetry of the wave-functions on each node of the
honeycomb. The chirality of the wave-functions with respect to a pseudo-spin
is also demonstrated for both Dirac cones Castro Neto  _et al._ (2009). This
rich electronic structure is attributed to the absence of hybridisation
between $1S$ and $1P$ bands. We show that the physics for fermions in
honeycomb optical lattices of cold atoms with $p$ orbitals Wu  _et al._
(2007); Sun  _et al._ (2011) could be studied in nanostructured 2D
semiconductors. We point out subtle differences between the electronic
structure of graphene- and silicene-type honeycomb structures. In the latter
case, gaps at the Dirac points can be controllably opened and closed by an
electric field applied perpendicularly to the 2D structure. In the valence
band of CdSe sheets, we demonstrate the existence of topological edge states
in the electronic gaps opened by the spin-orbit coupling and the quantum
confinement, supporting the recent work of Sushkov et al. using envelope-
function theory Sushkov and Castro Neto (2013). Our atomistic calculations
even predict multiple gaps with edge states.

##  II Geometry of the 2D lattices

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 3 of the 12 $\\{110\\}$ facets (light
green/orange) corresponding to a graphene-type honeycomb structure (both sub-
lattices in one plane). (b) Honeycomb lattice formed by attachment of 3 of the
8 $\\{111\\}$ facets (dark green/red), leading to the silicene-type
configuration (each sub-lattice in a different plane). In both models, a
$\langle 111\rangle$ 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 (Pb atoms: grey; Se atoms: yellow).

The 2D crystals that we consider in the present work are inspired by recent
experiments Evers  _et al._ (2013) discussed in Sec. X. These experiments
consist in the synthesis in a 2D reactor plane of honeycomb sheets of PbSe by
self-assembly and atomic attachment of (nearly) monodisperse PbSe colloidal
nanocrystals with a truncated cubic shape Dong  _et al._ (2010); Evers  _et
al._ (2013). Due to facet-specific atomic bonding, atomically-coherent
lattices with a honeycomb geometry and long-range periodicity are formed.
Rock-salt PbSe lattices are transformed into zinc-blende CdSe lattices by
cation exchange chemistry Son  _et al._ (2004); Evers  _et al._ (2013). In
both cases, the $\langle 111\rangle$ axis of the atomic lattice is
perpendicular to the plane of the honeycomb sheet.

The single-crystalline sheets that we consider are made of nanocrystals
arranged in honeycomb structure, as shown in Fig. 1. They can be seen as
triangular lattices with a basis of two nanocrystals per unit cell forming by
periodicity sub-lattices A and B. There are only two ways to assemble
truncated nanocubes with a $\langle 111\rangle$ body diagonal upright into a
honeycomb lattice. The first one is to use three $\\{110\\}$ facets
perpendicular to the superlattice plane with angles of 120 degrees, resulting
in the honeycomb lattices presented in Fig. 1a. In this structure all
nanocrystal units are organized in one plane, i.e. equivalent to a graphene-
type honeycomb lattice. In the second configuration, three truncated
$\\{111\\}$ planes per nanocrystal are used for atomic contact. This results
in atomically crystalline structures in which the nanocrystals of the A and B
sub-lattices are centred on different parallel planes, in analogy with atomic
silicene (Fig. 1b). The separation between the two planes is
$d/(2\sqrt{6})\approx 0.2d$ where $d$ is the superlattice parameter defined in
Fig. 1b. The unit cell of a typical graphene-type honeycomb lattice of PbSe
nanocrystals is displayed in Fig. 1c, in the case where polar $\\{111\\}$
facets are terminated by Pb atoms. We have also considered nanocrystals with
Se-terminated $\\{111\\}$ facets. Further details on the lattice geometry and
nanocrystal shape related to the degree of truncation are given in the
Appendix A.

The band structures presented below have been calculated for these quite
specific geometries inspired by experiments. However, very similar results can
be obtained for other honeycomb nano-geometries, as shown in Appendix C for
honeycomb lattices composed of spherical nanocrystals connected by cylindrical
bridges.

##  III Tight-binding methodology

Based on the effective-mass approach, Dirac-type bands of considerable width
can be expected in a nanoscale honeycomb lattice Park and Louie (2009).
However, a detailed understanding of the electronic structure in relation to
the atomic structure and nanoscale geometry of these systems requires more
advanced calculation methods. We have, therefore, calculated the energy bands
of honeycomb lattices using an atomistic tight-binding method. Each atom in
the lattice is described by a double set of $sp^{3}d^{5}s^{\star}$ atomic
orbitals including the spin degree of freedom. We include spin-orbit
interaction and we use tight-binding parameterizations (Appendix B) that give
very accurate band structures for bulk PbSe and CdSe. To avoid surface states,
CdSe structures are saturated by pseudo-hydrogen atoms. This is not necessary
in rock-salt PbSe structures as discussed in Ref. Allan and Delerue (2004).
Due to the large size of the systems that we have studied (up to $6\times
10^{4}$ atoms and $1.2\times 10^{6}$ atomic orbitals per unit cell), only
near-gap eigenstates are calculated using the numerical methods described by
Niquet et al. Niquet  _et al._ (2000).

##  IV Conduction band structure of graphene-type lattices of CdSe

Figure 2: Lowest conduction bands (a) and highest valence bands (b) 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
(VB), between the two 1S Dirac conduction bands (1S), and between the two 1P
Dirac conduction bands (1P) calculated at ${\bf K}+{\bf q}$ and plotted using
polar coordinates, i.e., energy versus the angle between ${\bf K}$ and ${\bf
q}$ (squares, red curve: $|{\bf q}|=0.05|{\bf K}|$ ; circles, blue curve:
$|{\bf q}|=0.15|{\bf K}|$ ; triangles, black curve: $|{\bf q}|=0.25|{\bf
K}|$).

The typical dispersion $[E({\bf k})]$ of the highest occupied bands and the
lowest unoccupied bands of a graphene-type superlattice of CdSe is shown in
Fig. 2. The electronic structure is composed of a succession of bands and gaps
due to the nanoscale periodicity. The honeycomb geometry induces periodic
scattering of the electronic waves, opening gaps in particular at the center
and at the edges of the superlattice Brillouin zone.

We first discuss the simpler but extremely rich physics of the lowest
conduction bands (Fig. 2a), which consist of two well-separated manifolds of
two and six bands (four and twelve bands including spin). Strikingly, the two
lowest bands have the same type of dispersion as the $\pi$ and $\pi^{\star}$
bands in real graphene; these bands are connected just at the K (${\bf k}={\bf
K}$) and K’ points of the Brillouin zone, where the dispersion is linear
(Dirac points). Moreover, in the second manifold higher in energy, four bands
have a small dispersion and two others form very dispersive Dirac bands. The
presence of two separated Dirac bands with Dirac points which can be
experimentally accessed by the Fermi level is remarkable and has not been
found in any other solid-state system. It can be understood from the
electronic structure of individual CdSe nanocrystals characterized by a spin-
degenerate electron state with a $1S$ envelope wave-function and by three
spin-degenerate $1P$ excited states higher in energy. The two manifolds of
bands arise from the inter-nanocrystal coupling between these $1S$ and $1P$
states, respectively. Interestingly, with this nanoscale geometry, the
coupling between nanocrystal wave-functions is strong enough to form
dispersive bands with high velocity at the Dirac points, but small enough to
avoid mixing (hybridization) between $1S$ and $1P$ states.

The systematic presence of nearly flat $1P$ bands is another remarkable
consequence of the absence of $S-P$ hybridization. The existence of
dispersionless bands has been predicted in honeycomb optical lattices of cold
atoms with $p$ orbitals Wu  _et al._ (2007); Sun  _et al._ (2011). In our
case, two $1P$ bands are built from the $1P_{z}$ states perpendicular to the
lattice, they are not very dispersive simply because $1P_{z}$-$1P_{z}$ ($\pi$)
interactions are weak. Two other $1P$ bands ($1P_{x,y}$), respectively above
and below the $1P$ Dirac band, are flat due to destructive interferences of
electron hopping induced by the honeycomb geometry Wu  _et al._ (2007); Sun
_et al._ (2011) (non-trivial flat bands).

Close to the Dirac points, for ${\bf k}={\bf K}+{\bf q}$ where $|{\bf
q}|<0.1|{\bf K}|$, the dispersion of the 1S and 1P Dirac bands is remarkably
isotropic, i.e., it does not depend on the angle between ${\bf q}$ and ${\bf
K}$ (Fig. 2c). For larger values of $|{\bf q}|$, the Dirac cones exhibit
trigonal warping due to the effect of the superlattice potential on the
electrons. Trigonal deformation of the energy bands has a profound impact on
the (quantum) Hall effect, interference patterns and weak localization in
graphite Dresselhaus (1974) and bilayer graphene Kechedzhi  _et al._ (2007).
Unusual phenomena such as enhanced interference around defects and
magnetically ordered exotic surfaces are also predicted at the surface of 3D
topological insulators due to the hexagonal warping of the bands Fu (2009);
Hasan  _et al._ (2009). By analogy, similar effects could arise in the
honeycomb superlattices of nanocrystals.

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 ($N_{\rm
at}$). 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).

At $\Gamma$ (${\bf k}=0$), the width of the $1S$ band increases not only with
decreasing nanocrystal size but also with increasing number of contact atoms
(Fig. 3). This can be understood by the fact that the contact area determines
the electronic coupling between the nanocrystal wave-functions of adjacent
sites, and acts in a similar way as the hopping parameter in atomic honeycomb
lattices (see Sec. IX). We predict a bandwidth above 100 meV for realistic
configurations. Note that the width of the Dirac $1P$ bands is even
considerably larger than that of the $1S$ bands. In the case of the graphene
geometry, we have found that the $1S$ and $1P$ bands are always characterized
by well-defined Dirac points.

Conduction bands at higher energy ($>2.6$ eV) in Fig. 2a are derived from the
$1D$ wave-functions of the nanocrystals. Interestingly, they also present flat
bands induced by the honeycomb geometry.

##  V Non-trivial gaps in the valence band of graphene-type lattices of CdSe

The twofold-degenerate conduction band of bulk CdSe near the $\Gamma$ point is
mainly derived from $s$ atomic orbitals and is therefore characterized by a
very weak spin-orbit coupling. It is the reason why $1S$ and $1P$ bands of
CdSe superlattices are spin-degenerate and exhibit well-defined Dirac points.
The situation is totally different in the valence bands of CdSe which are
built from $p_{3/2}-p_{1/2}$ atomic orbitals characterized by a strong spin-
orbit coupling, leading in the bulk to a splitting of 0.39 eV between heavy-
hole and split-off bands at $\Gamma$.

Therefore graphene-type lattices of CdSe nanocrystals present two extremely
interesting features when they are combined, 1) a honeycomb geometry, 2) a
strong spin-orbit coupling, here in the valence band. In this kind of systems,
the spin-orbit coupling may open a non-trivial gap and give rise to
topological insulators. These systems exhibit remarkable properties at their
boundaries characterized by helical edge states in the gap induced by the
spin-orbit coupling Hasan and Kane (2010); Qi and Zhang (2011); Beugeling  _et
al._ (2012). The edge states carry dissipation-less currents, leading in 2D
systems to quantum spin Hall effect, as initially predicted for graphene by
Kane and Mele Kane and Mele (2005). Whereas in graphene the spin-orbit
coupling is too small to give measurable effects Castro Neto  _et al._ (2009),
it was predicted that topological insulators with much larger gaps could be
made from ordinary semiconductors on which a potential with hexagonal symmetry
is superimposed Sushkov and Castro Neto (2013). We show below that honeycomb
lattices of CdSe nanocrystals actually present several non-trivial gaps in
their valence band.

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 width = 57 nm, periodic cell length $l=8.2$ nm). (b)
Ribbon with armchair edges (ribbon width = 33 nm, periodic cell length
$l=14.2$ nm). In each case, the unit cell is composed of sixteen nanocrystals
(34768 atoms per unit cell). The coloured regions indicate the bands of the
corresponding 2D semiconductor, i.e., those shown in Fig. 2b. 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.4a (energy of the states
$\approx-0.123$ 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.

The dispersion of the valence bands in CdSe superlattices (Fig. 2b) is much
more complex than the dispersion of conduction bands because anisotropic
heavy-hole, light-hole and split-off bands are coupled by the confinement.
Dirac points are not visible in spite of the honeycomb geometry. However, the
highest valence bands in Fig. 2b roughly behave like the $\pi-\pi^{\star}$
bands in graphene, their energy-momentum relationship is isotropic close to
the K point (Fig. 2c), but there is a large gap between the two bands. Other
gaps are present at lower energy.

In the following, we investigate the topological properties of these gaps.
First, we calculate the Chern numbers for the two highest valence bands of the
2D sheet Hatsugai (1993). The methodology and the results are described in
Appendix D. The analysis of these Chern numbers demonstrates the non-trivial
character of the two gaps between the highest valence bands and therefore the
quantum spin Hall effect is predicted in these gaps. Second, we calculate the
valence band structure of ribbons with zigzag and armchair geometry built from
CdSe nanocrystals. We have considered the same honeycomb geometry as in Fig.
2b but for a 1D ribbon instead of a 2D sheet. The unit cell forming the ribbon
by periodicity is composed of sixteen nanocrystals. Figure 4a presents the 1D
band structure for a ribbon with zigzag edges. We have found edge states
crossing the gap between the two highest valence bands of the 2D sheet. The 2D
plot of their wave-functions is shown in Fig. 5. Interestingly, edge states
are also present in the second gap below in energy, and others are even
visible in the smaller gaps below -0.15 eV. The analysis of their wave-
functions on one side of the ribbon shows that the spin is mainly oriented
perpendicular to the lattice ($>98\%$) Not , and that the direction of the
spin is reversed for motion in the opposite direction ($k\to-k$). The
situation is inverted at the opposite edge of the ribbon. Very similar results
are obtained for ribbons with armchair edges (Fig. 4b) and for ribbons in
which we modify the size of the nanocrystals at the edges (not shown). The
presence of helical edge states in the gaps of the 2D sheet and their
robustness with respect to the edge geometry are signatures of their non-
trivial topology. The multiplicity of non-trivial gaps demonstrates the
variety of effects induced by the honeycomb nano-geometry on the band
structure of the 2D semiconductor, contrarily to the case of a square nano-
geometry Kalesaki  _et al._ (2013).

Figure 6: Evolution of the energy gap between the two highest valence bands of
CdSe sheets with graphene-like honeycomb geometry as a function of the number
of atoms that form the nanocrystal/nanocrystal contact ($N_{\rm at}$). The
size of the nanocrystals is indicated by the body diagonal of the truncated
nanocube.

The gaps between the valence bands are tunable thanks to the quantum
confinement. Figure 6 shows that the gap between the two highest valence bands
strongly depends on the size of the nanocrystals and increases with the number
of atoms at the contact plane between neighbor nanocrystals. A gap above 10
meV is possible with nanocrystal size below 4 nm.

##  VI Pseudo-spin

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. 2a). The
phase shift is calculated at ${\bf K}+{\bf q}_{1}$ and ${\bf K}+{\bf q}_{2}$
with $|{\bf q}_{1}|=|{\bf q}_{2}|=0.05|{\bf K}|$ and an angle between ${\bf
q}_{1}$ and ${\bf q}_{2}$ of 3.8 radians, 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 ${\bf q}_{1}$ and
${\bf q}_{2}$, at K (blue, square symbols) and K’ (red, circular symbols).
Similar results are obtained for the $1P$ Dirac cone.

A fingerprint of the electronic states at the Dirac cones in graphene is their
chirality with respect to a pseudo-spin associated with the two components of
the wave-function on the two atoms of the unit cell Castro Neto  _et al._
(2009). Figure 7 shows that the pseudo-spin is also well defined near the
Dirac points in the conduction band of CdSe honeycomb lattices, in spite of
the fact that each unit cell of the structure contains thousands of atoms.
When we rotate the ${\bf k}$ vector around the K point, the phase shift of the
wave-function is almost constant across each nanocrystal, in other words, it
does not depend on the atomic orbital and its position. Note that the wave-
function phase shift changes quite abruptly at the contact plane between two
nanocrystals (one of sub-lattice A and one of sub-lattice B). In the lower
$1S$ band, the phase difference between nanocrystals A and B is equal to the
angle of rotation, and the variation is opposite at the K’ point (Fig. 7b).
The sign is also inverted in the upper $1S$ band. This chirality of the wave-
function should reduce backscattering of the Dirac electrons, for the same
reasons as in graphene Castro Neto  _et al._ (2009).

##  VII Band structure of silicene-type lattices of CdSe

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.

The band structures for honeycomb lattices of CdSe with silicene-type geometry
are almost the same as for the graphene-type geometry (see Fig. 8a for
conduction bands), but there are gaps at the Dirac points due to the absence
of mirror symmetry with respect to the $\\{111\\}$ contact plane between
neighbour nanocrystals. In other words, sub-lattice A is not equivalent to
sub-lattice B. The gap at K strongly depends on the size and the shape of the
nanocrystals. Since in that case the nanocrystals of the A and B sub-lattices
are positioned at different heights, the electronic structure is very
sensitive to an electric field applied along $\langle 111\rangle$ as shown in
Fig 8b. The gap at K between the $1S$ bands varies linearly with the field and
vanishes when the potential drop between nanocrystals A and B compensates the
effect of the geometrical asymmetry on the $1S$ wave-functions. The variation
of the gap at K between $1P$ Dirac bands is more complex due to intermixing
between $1P$ nanocrystal wave-functions, and increasing $1S-1P$ and $1P-1D$
hybridization. However, this gap tends to zero at increasing electric field.

##  VIII Band structure of graphene-type lattices of PbSe

Figure 9: Lowest conduction bands (a) and highest valence bands (b) of a
graphene-type honeycomb lattice of truncated nanocubes of PbSe (body diagonal
of 5.00 nm, $q=0.5$).

Figure 9 shows a typical band structure of a honeycomb lattice of PbSe with
graphene-type geometry. The lowest conduction bands and highest valence bands
are characterized by a manifold of eight bands which are formed by the
$2\times 4$ $1S$ conduction and valence wave-functions of the two PbSe
nanocrystals of the unit cell. These $1S$ nanocrystal wave-functions are
derived from the fourfold degenerate conduction and valence bands of bulk PbSe
at the L point of the Brillouin zone (eightfold degeneracy if we include the
spin). The conduction bands at higher energy and the valence bands at lower
energy are derived from the 24 $1P$ nanocrystal wave-functions (only seven
bands are shown for clarity). The complex dispersion of all these bands, even
the $1S$ ones, shows that the coupling between the wave-functions of neighbor
nanocrystals is not only governed by the symmetry of the envelope function
($1S$, $1P$) defined by the nano-geometry but also, in a subtle way, by the
underlying Bloch function which depends on the originating valley. These
results illustrate the fact that both the atomic lattice of the parent
semiconductor and the overall nano-geometry influence the band structure.

##  IX Effective tight-binding model

The overall behavior of the $1S$ and $1P$ conduction bands of honeycomb
lattices of CdSe nanocrystals can be interpreted using a simple (effective)
tight-binding Hamiltonian of graphene or silicene in which the two ”atoms” of
the unit cell are described by one $s$ and three $p$ orbitals (doubled when
the spin is considered). The on-site energies are $E_{s}(A)$,
$E_{p_{x}}(A)=E_{p_{y}}(A)$, $E_{p_{z}}(A)$, and $E_{s}(B)$,
$E_{p_{x}}(B)=E_{p_{y}}(B)$, $E_{p_{z}}(B)$ for the A and B sub-lattices,
respectively. The $z$ axis is taken perpendicular to the lattice. For
silicene, sub-lattice B is not in the same plane as A, the angle between the
AB bond and the $z$ axis is taken to be the angle between the $[111]$ and
$[11-1]$ crystallographic axes. The energy of the $p_{z}$ orbital is allowed
to be different from the $p_{x}$ and $p_{y}$ orbitals. Following Slater and
Koster Slater and Koster (1954), all nearest-neighbor interactions (hopping
terms) can be written in the two-center approximation as functions of four
parameters ($V_{ss\sigma}$, $V_{sp\sigma}$, $V_{pp\sigma}$, $V_{pp\pi}$) plus
geometrical factors. The problems for the $s$ and $p$ bands are separable when
$V_{sp\sigma}=0$. We have found that the results of full tight-binding
calculations for all 2D crystals considered are only compatible with small
values of $V_{sp\sigma}$, i.e., by a small hybridization between $s$ and $p$
orbitals.

Figure 10: Band structure calculated for a silicene-type lattice using the
effective tight-binding Hamiltonian for three sets of parameters. (a)
$E_{s}(A,B)=2.18$ eV, all the $E_{p}=2.44$ eV, $V_{ss\sigma}=-8$ meV,
$V_{sp\sigma}=0$ meV, $V_{pp\sigma}=50$ meV, $V_{pp\pi}=-2$ meV. (b) Same as
(a) but with $E_{p_{z}}(A,B)=2.51$ eV. (c) $E_{s}(A)=2.18$ eV, $E_{s}(B)=2.19$
eV, $E_{p_{x}}(A)=E_{p_{y}}(A)=2.44$ eV, $E_{p_{x}}(B)=E_{p_{y}}(B)=2.46$ eV,
$E_{p_{z}}(A)=2.51$ eV, $E_{p_{z}}(B)=2.52$ eV, $V_{ss\sigma}=-8$ meV,
$V_{sp\sigma}=0$ meV, $V_{pp\sigma}=50$ meV, $V_{pp\pi}=-2$ meV.

Typical band structures for the silicene geometry are shown in Fig 10. Figure
10a corresponds to a case where the on-site energies are the same on the sub-
lattices A and B, and all $p$ orbitals have the same energy. Dirac points are
obtained at three energies, one in the $s$ bands, two in the $p$, and no gap
formed at K. This situation has not been found for more realistic band
structure calculations. One Dirac $p$ band is suppressed when we slightly
shift the $p_{z}$ energy with respect to the $p_{x}$, $p_{y}$ energies (Fig
10b). Finally, a small gap is opened at the two Dirac points when the orbital
energies in sub-lattice A are not the same as in B (Fig 10c). Figure 10c is in
close agreement with Fig. 8a highlighting the suitability of the present
effective Hamiltonian to describe the obtained results.

##  X Experimental perspective

Several methods can be proposed to confine semiconductor carriers in a
honeycomb geometry. The simplest method would be to use a hexagonal or
honeycomb array of a metal acting as a geometric gate that would
electrostatically force the carriers in a honeycomb lattice. Theoretical and
experimental work with GaAs/GaAlAs has been recently reported Gibertini  _et
al._ (2009); Simoni  _et al._ (2010); Sushkov and Castro Neto (2013). The
advantage of this approach is the substantial theoretical and experimental
knowledge already attained in the field. However, due to the relatively large
period of the honeycomb lattice, the width of the bands is limited to a few
meV Park and Louie (2009). A second method would be to prepare suitable
templates by lithography, and use these to grow semiconductor lattices with
wet-chemical or gas-phase methods (chemical vapor deposition and molecular
beam epitaxy). General concerns relate to the crystallinity of the 2D system
prepared with a honeycomb geometry. Our tight-binding calculations
unambiguously show that the electronic structure is determined both by the
periodicity of the 2D semiconductor in the nanometer range and the atomic
lattice. Recently, a third approach to prepare semiconductors with a honeycomb
geometry came from a rather unexpected corner. Evers et al. showed that
atomically-coherent sheets of a PbSe semiconductor can be prepared by oriented
attachment of colloidal nanocubes Evers  _et al._ (2013). Moreover, the as-
prepared systems can be transformed into zinc-blende CdSe by Cd-for-Pb ion
exchange, preserving the nano-geometry Bon . The systems showed astounding
geometrical order and a well-defined atomic structure. A challenge with theses
systems will be to incorporate them in a field-effect transistor, such that
the transport properties can be measured. As with the template-approach, the
effect of defects in the atomic lattice and disorder in the superimposed
honeycomb geometry, electronic doping, and the surface termination of the 2D
honeycomb semiconductors are issues that need to be further addressed.

##  XI Summary and future directions

In conclusion, we have shown that atomically-coherent honeycomb lattices of
semiconductors with period below 10 nm present very interesting band
structures, which are defined both by the properties of the parent
semiconductor and the nano-geometry. We have calculated the electronic
structure of these materials using an atomistic tight-binding method,
considering graphene- and silicene-type lattices obtained by the assembling of
semiconductor nanocrystals. In the case of honeycomb lattices of CdSe, we
predict a rich conduction band structure exhibiting non-trivial flat bands and
Dirac cones at two experimentally reachable energies. These bands are derived
from the coupling between nearest-neighbor nanocrystal wave-functions with
$1S$ and $1P$ symmetry. The formation of distinct Dirac cones is possible
because of the weak hybridization between $1S$ and $1P$ wave-functions under
the effect of the strong quantum confinement. We also predict the opening of
non-trivial gaps in the valence band of CdSe sheets due to the effect of the
nano-geometry and the spin-orbit coupling. Several topological edge states are
found in these gaps. The possibility to have multiple Dirac cones, non-trivial
flat bands and topological insulating gaps in the same system is remarkable.

Recent experiments strongly suggest that the synthesis of such single-
crystalline sheets of semiconductors is possible using facet-specific
attachement of nanocrystals Evers  _et al._ (2013). Therefore, numerous
directions are open for the theoretical and experimental investigation of
these systems. The electronic structure and carrier transport can be studied
by local scanning tunnelling microscopy and spectroscopy, and in a field-
effect transistor. With illumination and/or gating the conduction band can be
controllably filled with electrons up to several electrons per nanocrystal
Roest  _et al._ (2002); Yu  _et al._ (2003); Talapin and Murray (2005),
allowing for the Fermi level to cross the Dirac points. Our work provides
evidence for non-trivial flat $1P$ bands, and the lowest one can be reached at
a nanocrystal filling between 2 and 3 electrons. Electron-electron
interactions should then play a crucial role and may lead to Wigner
crystallization Wu  _et al._ (2007). We remark that the physics of honeycomb
lattices of $p$ orbitals is largely unexplored. It will be particularly
attractive to consider honeycomb lattices of semiconductors with even stronger
spin-orbit coupling, in which we expect the emergence of new electronic
(topological) phases.

Finally, it is important to realize that all these interesting properties
(Dirac cones, non-trivial flat bands and topological edge states) originate
from the honeycomb geometry. For instance, the band structures of atomically-
coherent square semiconductor superlattices that we discussed recently
Kalesaki  _et al._ (2013), although of interest in their own respect, do not
contain any of the Dirac-based quantum electronic properties of similar
semiconductors with a honeycomb nano-geometry, discussed here.

###### Acknowledgements.

This work has been supported by funding of the French National Research Agency
[ANR, (ANR-09-BLAN-0421-01)], Netherlands Organisation for Scientific Research
(NWO) and FOM [“Control over Functional Nanoparticle Solids” (FNPS)]. C.D. was
visiting professor at the Debye Institute for Nanomaterials Science, Utrecht
University, at the time of this research.

##  Appendix A Details on the geometry of the honeycomb lattices

###  A.1 Formation of the nanocrystals

The nanocrystals are built from PbSe nanocubes on which six $\\{100\\}$, eight
$\\{111\\}$ and twelve $\\{110\\}$ facets are created by truncation. The
positions of the vertices of the nanocrystal shape are given by $P[\pm
1,\pm(1-q),\pm(1-q)]$ where $[\pm 1,\pm 1,\pm 1]$ indicate the position of the
six corners of the original nanocube, $q$ is the truncation factor, and $P$
represents all the possible permutations. We have considered realistic shapes
corresponding to $q$ between 0.25 and 0.5 Evers  _et al._ (2013). The
truncated $\langle 111\rangle$-oriented nanocubes are assembled into two types
of honeycomb lattices with a graphene-like or silicene-like shape as described
below.

###  A.2 Geometry of graphene-type supercells

Two nanocrystals (A and B) are attached along the $\langle 110\rangle$
direction (perpendicular to the $\langle 111\rangle$ axis) to define a unit
cell, forming by periodicity an atomically-coherent honeycomb lattice. Each
nanocrystal is defined by the same number of bi-planes of atoms in the
$\langle 110\rangle$ direction but there is an additional plane of atoms
shared between neighboring nanocrystals in order to avoid the formation of
wrong bonds. The length of the two vectors defining the superlattice is
$(2n+1)a\sqrt{(}6)/2$ where $n$ is an integer and $a$ is the lattice parameter
(0.612 nm for PbSe, 0.608 nm for CdSe). Since the atomistic reconstruction at
the contact plane between neighboring nanocrystals is not precisely known, we
have also considered structures in which we have slightly enlarged each inter-
nanocrystal junction by one line of atoms on each side of the $\\{110\\}$
facets.

###  A.3 Geometry of silicene-type supercells

The two nanocrystals A and B are attached along a $\\{111\\}$ facet to define
a unit cell and an atomically-coherent honeycomb lattice is formed by
periodicity. Note however that the A and B nanocrystal sub-lattices are each
located in a different plane. The length of the superlattice vectors
(superlattice parameter) is $d=ma/\sqrt{2}$ where $m$ is an integer. The
separation between the two planes A and B is $ma/(4\sqrt{3})$. Similar to the
graphene-type lattices, we have also considered structures in which the
$\\{111\\}$ facets at the contact plane between neighboring nanocrystals are
enlarged.

##  Appendix B Tight-binding parameters

Table 1: Tight-binding parameters (notations of Slater and Koster Slater and
Koster (1954)) for zinc-blende CdSe in an orthogonal $sp^{3}d^{5}s^{\star}$
model. $\Delta$ is the spin-orbit coupling. (a) and (c) denote the anion (Se)
and the cation (Cd), respectively. Parameters for CdSe (eV)
---
$E_{s}(\rm a)$ | -8.065657 | $E_{s}(\rm c)$ | -1.857148
$E_{p}(\rm a)$ | 4.870028 | $E_{p}(\rm c)$ | 5.613460
$E_{d_{xy}}(\rm a)$ | 15.671502 | $E_{d_{xy}}(\rm c)$ | 16.715749
$E_{d_{x^{2}-y^{2}}}(\rm a)$ | 15.232107 | $E_{d_{x^{2}-y^{2}}}(\rm c)$ | 20.151047
$E_{s^{*}}(\rm a)$ | 15.636238 | $E_{s^{*}}(\rm c)$ | 20.004452
$\Delta(\rm a)$ | 0.140000 | $\Delta(\rm c)$ | 0.150000
$V_{ss\sigma}({\rm ac})$ | -1.639722 | $V_{s^{*}s^{*}\sigma}({\rm ac})$ | -1.805116
$V_{ss^{*}\sigma}({\rm ac})$ | 1.317093 | $V_{ss^{*}\sigma}({\rm ca})$ | 0.039842
$V_{sp\sigma}({\rm ac})$ | 3.668731 | $V_{sp\sigma}({\rm ca})$ | 1.885956
$V_{s^{*}p\sigma}({\rm ac})$ | 0.978722 | $V_{s^{*}p\sigma}({\rm ca})$ | 1.424094
$V_{sd\sigma}({\rm ac})$ | -0.890315 | $V_{sd\sigma}({\rm ca})$ | -1.007270
$V_{s^{*}d\sigma}({\rm ac})$ | 0.906630 | $V_{s^{*}d\sigma}({\rm ca})$ | 2.472941
$V_{pp\sigma}({\rm ac})$ | 4.430196 | $V_{pp\pi}({\rm ac})$ | -0.798156
$V_{pd\sigma}({\rm ac})$ | -2.645560 | $V_{pd\sigma}({\rm ca})$ | -1.296749
$V_{pd\pi}({\rm ac})$ | 0.028089 | $V_{pd\pi}({\rm ca})$ | 2.295717
$V_{dd\sigma}({\rm ac})$ | -2.480060 | $V_{dd\pi}({\rm ac})$ | 2.393224
$V_{dd\delta}$ | -1.373199 |  |
Parameters for Cd-H and Se-H (eV)
$E_{H}$ | 0.000000 |  |
$V_{ss\sigma}$ | -35.69727 | $V_{sp\sigma}$ | 61.82948

We consider a double basis of $sp^{3}d^{5}s^{\star}$ atomic orbitals for each
Pb, Cd or Se atom, including the spin degree of freedom. For PbSe, we use the
tight-binding parameters as given in Ref. Allan and Delerue (2004). Due to the
lack of $sp^{3}d^{5}s^{\star}$ tight-binding parameters for zinc-blende CdSe,
corresponding data were derived and are presented in Table 1. The use of a
$sp^{3}d^{5}s^{\star}$ basis allows us to get a very reliable band structure
for bulk materials compared to ab initio calculations and available
experimental data: see Ref. Allan and Delerue (2004) for PbSe. Hence, these
parameters can be safely transferred to predict the electronic structure of
semiconductor nanostructures Delerue and Lannoo (2004).

The effects of the electric field on the electronic structure are calculated
assuming that the field inside the superlattice is uniform. In this
approximation, the calculation of the screening in such a complex system is
avoided. The main conclusions of these calculations, i.e., the strong
evolution of the bands with the applied field and the possibility to close or
open the gap at the $1S$ Dirac point in silicene-type lattices, do not depend
on this approximation.

##  Appendix C Results for another type of honeycomb nano-geometry

Figure 11: Lowest conduction bands of a graphene-type honeycomb lattice of
spherical nanocrystals of CdSe (diameter = 4.7 nm). Nearest-neighbor
nanocrystals are connected by cylindrical bridges of CdSe (diameter of the
cylinders = 2.3 nm).

It is also possible to build a honeycomb lattice using spherical nanocrystals.
We set the distance between nearest-neighbor nanocrystals equal to the
diameter, i.e., the spheres are tangential. Between each pair of neighbors, we
add a cylinder of atoms which serves as a bond. We predict that the electronic
bands for these 2D crystalline sheets are very close to those obtained with
truncated nanocubes. A typical result for the conduction band is shown in Fig.
11 where the two Dirac cones and the flat bands are present. We have found
similar band structures for diameters of the cylinders up to 80% of the
diameter of the spheres, demonstrating the robustness of the results. The
width of the Dirac bands is related to the strength of inter-nanocrystal
bonds, i.e., to the diameter of the cylinders. The non-trivial flat bands are
easily identified in Fig. 11 since they are connected to the $1P$ Dirac band.
Similar results are also obtained for silicene-type geometry (not shown).

##  Appendix D Calculation of the Chern numbers

We have calculated the Chern numbers of the highest valence bands following
the methodology proposed by Fukui et al. Fukui  _et al._ (2005). For each band
$n$, we calculate the wave-functions $|n,{\bf k}\rangle$ on a $N\times N$ grid
within the Brillouin zone, i.e., ${\bf k}=m_{1}{\bf a}_{1}^{*}/N+m_{2}{\bf
a}_{2}^{*}/N$ where ${\bf a}_{1}^{*}$ and ${\bf a}_{2}^{*}$ are the reciprocal
lattice vectors ($m_{1},m_{2}=0,...,N-1$). In order to specify the gauge, we
make the transformation $|n,{\bf k}\rangle\to|\langle n,{\bf
k}|0\rangle|^{-1}\langle n,{\bf k}|0\rangle|n,{\bf k}\rangle$ where
$|0\rangle$ is an arbitrary state which was chosen as constant over the unit
cell. The lattice Chern number associated with the $n$th band is calculated as

$\tilde{c}_{n}=\frac{1}{2\pi i}\sum_{{\bf k}}\tilde{F}({\bf k})$ (1)

where $\tilde{F}({\bf k})$ is defined as

$\tilde{F}({\bf k})=\ln\left[U_{1}({\bf k})U_{2}\left({\bf k}+\frac{{\bf
a}_{1}^{*}}{N}\right)U_{1}^{-1}\left({\bf k}+\frac{{\bf
a}_{2}^{*}}{N}\right)U_{2}^{-1}({\bf k})\right]$ (2)

with $U_{i}({\bf k})=\langle n,{\bf k}|n,{\bf k}+{\bf
a}_{i}^{*}/N\rangle/\left|\langle n,{\bf k}|n,{\bf k}+{\bf
a}_{i}^{*}/N\rangle\right|$. $\tilde{F}({\bf k})$ is defined within the
principal branch of the logarithm ($-\pi<\tilde{F}({\bf k})/i\leq\pi$). It was
shown that the lattice Chern number $\tilde{c}_{n}$ tends toward the usual
Chern number $c_{n}$ in the limit $N\to\infty$ Fukui  _et al._ (2005).

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
${\bf B}$ which induces Zeeman splitting of the bands ($\mu_{B}g_{S}B=1$ meV).
The corresponding bands at zero magnetic field are those shown in Fig. 2b. The
Chern numbers of the bands are indicated on the figure.

In the calculation of the Chern numbers for the superlattices, numerical and
fundamental problems may arise. Numerical problems come from the size of the
systems containing thousands of atoms per cell. However, the calculation of
the quantities $\tilde{F}({\bf k})$ can be done separately for each value of
${\bf k}$ on the grid, enabling powerful parallel treatment on multi-processor
computers. As a consequence, the methodology proposed by Fukui et al. Fukui
_et al._ (2005) was found to be extremely efficient.

Fundamental issues are more problematic. Each band among the highest valence
bands shown in Fig. 2b is actually composed of two almost-degenerate bands
which intersect at several points of the Brillouin zone ($\Gamma$, K, M in
particular). Elsewhere in the Brillouin zone, the splitting between the
almost-degenerate bands is smaller than 0.2 meV in average (not visible in
Fig. 2b). Therefore the methodology to calculate $\tilde{c}_{n}$ cannot be
applied to these situations as it requires non-degenerate bands over the full
Brillouin zone Fukui  _et al._ (2005). In order to lift the degeneracies
between the bands, we have studied the effect of a magnetic field applied
perpendicularly to the superlattices. Since our objective is to explore the
topological properties of the bands, we have only considered the Zeeman part
of the coupling Hamiltonian. Figure 12 shows that the application of a
magnetic field totally splits the two highest valence bands, pushing downward
(upward) the states with a majority of spin up (down) component.

The calculated Chern numbers for the two highest manifolds of valence bands
are indicated in Fig. 12. For each band, the sum in Eq. (1) typically
converges to its final value for $N\approx 30$. We have checked that
$\tilde{c}_{n}$ remains constant for larger $N$ from $\approx 30$ to $100$. It
also remains constant when we vary the magnetic field, from
$\mu_{B}g_{S}B\approx 0.2$ meV (minimum value to remove the degenaracies) to 5
meV. It was not possible to calculate the Chern numbers for the valence bands
lower in energy in this manner, because the degeneracy points remain even
under an applied magnetic field. An equivalent approach to characterize the Z2
topological invariant is given in Refs. Fu and Kane (2007); Fukui and Hatsugai
(2007).

For each valence band shown in Fig. 12, the sum of the Chern numbers
$c_{\uparrow,n}+c_{\downarrow,n}$ for the two spin components vanishes meaning
that their contribution to the Hall conductivity is zero. However, the
difference $c_{\uparrow,n}-c_{\downarrow,n}$ (spin Chern number) which is
linked to the spin Hall conductivity Beugeling  _et al._ (2012) does not
vanish. Since the sum of the Chern numbers over all the occupied valence bands
must be zero for each spin component (there is no edge state in the gap
between valence and conduction bands), we deduce by subtraction that the
quantum spin Hall effect must be present in the two highest gaps of the
valence band, in total agreement with our previous conclusions of Sec. V.

A small remark is in place here. For the reason that the spins are not
pointing exactly up or down, the difference $c_{\uparrow,n}-c_{\downarrow,n}$
is strictly speaking not well defined. However, the spins are tilted only
slightly away from the vertical (deviation $<2\%$), so that they can be
treated as approximately up or down. The Chern numbers indicate that the
corresponding spin “up” and spin “down” edge states counterpropagate, i.e.,
that the bulk gap exhibits a nonzero spin Hall conductivity while the charge
Hall conductivity vanishes. The small tilt of the spins perturbs the value of
the spin Hall conductivity slightly away from the quantized value, but does
not destroy the topological character of the state. Thus, this state is a
quantum spin Hall state in good approximation.

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