Artificial graphene as a tunable Dirac material
Marco Polini,
1,
∗
Francisco Guinea,
2
Maciej Lewenstein,
3, 4
Hari C. Manoharan,
5, 6
and Vittorio Pellegrini
1
1
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
2
Instituto de Ciencia de Materiales de Madrid (CSIC ) ,
Sor Juana In´es de la Cruz 3, E-28049 Madrid, Spain
3
ICFO - Institut de Ci`encies Fot`oniques, Mediterranean Techn ol ogy Park,
Av. Carl Friedrich Gauss 3, E-08860 Castelldefels, Barcelona, Spain
4
ICREA - Instituci´o Catalana de Recerca i Estudis Avan¸ca ts, 08010 Barcelona, Spain
5
Department of Physics, Stanford University, Stanf ord, California 94305, USA
6
Stanford Institute for Materials and Energy Sciences,
SLAC National Accelerator Laboratory, Menlo Park, California 94025, U SA
Artificial honeycomb lattices off er a tunable platform to study massless Dirac quasiparticles and
their topological and correlated phases. Here we review recent progress in the design and fabrica-
tion of such synthetic structures focusing on nanopat tern i n g of two-dimensional electron gases in
semiconductors, molecule-by-molecule assembly by scanning probe methods, and optical trap p i n g
of ultracold atoms in crystals of light. We also discuss photonic crystals with Dirac cone dispersion
and topologically protected edge states. We emphasize how the interplay between singl e- p a rti c le
band structure engineering and cooperative effects leads to spectacular manifestations in tunneling
and optical spectroscopies.
INTRODUCTION
Graphene is boasting a profound impact in condensed-
matter science [
1–5] and technology [6–8]. It is an unusu-
ally perfect realization of a 2D semimetal displaying, in a
wide range of energies, linearly dispersing conduction and
valence bands, which touch at the so-called Dirac point.
Charge neutrality pins the Fermi energy at the apex of
the Dirac cones, which are particularly intriguing since
they imply quasiparticles that behave like relativistic el-
ementary particles with zero rest mass [
9]. Dirac cones
characterize other materials. For instance, they describe
the chiral conducting surface states th at emerge on the
surface of 3D topological insulators (TIs) [
10, 11].
In the case of Dirac materials, the standard analysis
of fermionic systems, based on concepts like “effective
mass” and energy gap, requires an extensive reformula-
tion. The pr op agat ion of Di rac fermions shows unusual
features [
9], traceable to the presence of a new physical
variable, the sublattice-pseudospin degree of freedom. In-
teractions between Dirac fermions [
12] are akin to those
studied in quantum electrodynamics (QED) and there-
fore very different from interactions between Schr¨odinger
electrons in ordinary metals and semiconductors.
Dirac conical singularities can actually emerge in any
2D lattice. Symmetry arguments, in fact, show that t h ey
should appear regularly at the corners of the Brillouin
zone (BZ) in lattices with triangular symmetry. There-
fore Dirac fermion physics and i t s technological ex p loi t a-
tion should not be considered restricted to the r eal m
of “natural” materials—such as graphene and TIs—but
could also b e explore d, in principle, in artificial structures
displaying t ri an gul ar symmetry. Advantages of such “ar -
tificial graphene” (AG) systems are likely to be th ei r tun-
able properties—including lattice con st ants, hopping en-
ergies, and inter-particle interaction strength—and the
possibility to design and realize artificial def ect s . Ad-
ditionally, the creat i on of such structur es is particularly
attractive for the study of quantum phases driven by cor-
relation effects, which in natural (single-layer) graphene
seem to emerge only in ultra-high magnetic fields [
12].
Recent advances have demonstrated the possibility of
creating AG in diverse subfields of low-energy p hysics.
Current methods of design and synthesis include i)
nanopatterning of ultra-high-mobility 2D electron gases
(EGs) [
13–18], ii) molecule-by-molecule assembly on
metal surfaces by scanning probe me t hods [
19], ii i) trap-
ping ultracold fermionic and bosonic atoms in honeycomb
optical lattices [
20–22], and iv) confining photons in hon-
eycomb photonic crystals [
23–26]. Figure 1 summarizes
these four different approaches and the relevant tunable
parameters. Here we rev i ew the state of the art of this
emerging multidisciplinary field and offer perspectives on
possible futur e developments.
DESIGNING AND PROBING DIRAC BANDS IN
ARTIFICIAL LATTICES
AG structures may not immediately compete with
graphene for technological applications. However, they
offer a playground to observe and comprehend physical
phenomena related to Dirac energy-momentum disper-
sion relations in regimes that are di fficu l t to achieve in
natural graphene. In the following we discuss the most
relevant AG lattices explored so far in which electrons,
photons, cold atoms, and i ons are the confin ed particles.
As we elaborate below, these systems have complemen-
tary physical properties that enable the investigation of
a wide range of phenomena.
arXiv:1304.0750v1 [cond-mat.mes-hall] 2 Apr 2013
2
FIG. 1: Artificial graphene structures experimentally obtained by different methods. (a) Sc a n n in g electron micro-
graph o f the surface of a nanopatterned GaAs heterostructure. This is achieved by defining first an array of Nickel disks with
the desired geometry by e-beam nanolithography and then by etching away the material outside the disks by inductive-coupled
reactive ion shallow etching and by Nickel removal. The bottom panel lists the main tunable parameters of e a ch approach: V
0
is the lattice depth, d is the lattice constant, and N the numb er of sites. The parameter U (V ) is the strength of the on-si te
(nearest-neighbor interaction) repulsion while t is the hopp i n g energy scale. Finally, T
F
is the Fermi temperature. Panels (b)
and (c) refer to molecular graphene systems an d optical lattices for cold atoms, respectively. In panel (b) red (black) spheres
represent the Oxygen (Carbon) atoms of CO molecules while the yellow/orange surfaces represent the electron density in a
honeycomb p a t te rn . Panel (d) describes photonic honeycomb crystals induced by optical induction methods. ∆n is the change
of refractive index induced by laser irradiation. The parameter c
0
is the coupling constant between waveguides, which plays
the role of the hopping parameter t. There is no Hubba rd gap or band gap in this system (only the one that is induced by th e
strain in the form of Landau levels). Panel (c) courtesy of L. Fallani.
Confining electrons
In 1970 Esaki and Tsu realized the possibility to en -
gineer energy bands by artificially modulating th e po-
tential in one direct i on [
27]. They came up with the
idea of using a semiconductor superlattice and predicted
the onset of negative differential conductivity. This pio-
neering work stimulated a large effort worldwide focused
on band-gap engineering in semiconductor heterostruc-
tures that, thanks to the r efi n eme nt of nanofabrication
techniques, eventually led to the development of lateral
superlattices—semiconductor systems characterized by a
2D periodi c potential modulation [
28].
The advent of these artifici al crystals with a tun-
able band-structure has greatly influ en ce d the field of
2DEGs in modulation-doped semicond uc tor heterostruc-
tures. Works in t h i s area started in the l at e eighties and
enabled the observation of Weiss oscillations [
29], novel
conductance resonances due to quantization of the elec-
tron orbits in the 2D pattern in a magnetic field [30],
chaotic dynamic s, and, more recently, led to studies of
Hofstadter butterfly phenomena [
31, 32]. Recently, simi-
lar effects have been observed in natural graphene where
the 2D periodic potential (with periodicity of the order
of ∼ 10 nm) was induced by placing i t on h-BN [
33–35].
The ever-increasing toolbox of nano-fabrication meth-
ods allows t oday a large flexibility in realizing high qual-
ity 2D patterns with nanoscale dimensions in semicon-
ductor quantum structures hosting ultra-high mobility
electrons. An external potenti al landscape with honey-
comb geometry that acts as a lattice of potential wells
(such as quantum dots) to trap electrons and/or holes can
be obtained by a combination of e-beam nano lithogra-
phy, reactive ion etching and deposition of metallic gates.
The spatial resolution of these techniques c an reach val-
ues of a few tens of nanometer or even below. Further
improvements in spatial resolution can b e obtain ed by
bottom-up nanofabrication methods, e.g. by designing
semiconductor lattices by nanocrystal self assembly [
36].
The possibility of independently controlling the electron
density and inter-site distances allows one to tune the in-
terplay betwe en on-site (U) and nearest-neighbor (V ) re-
pulsive interactions and single-particle hopping (t), open-
ing the way to the observation of collective phenomen a
and quantum phase transitions in such AG solid-state
systems.
Recent experimental results obtained in honeycomb
patterns defined on 2DEGs in GaAs quantum het-
3
erostructures offer ex ci t i n g evidence t hat AG in semi-
conductors can be realized in the laboratory [14–17].
Available experimental results in the re gi me U/t ≫ 1
reveal [
16] unique low-lying collective excitations, such
as “anomalous spin waves”, i n spectra of inelastic light
scattering (see also below and Fig.
5a-c). Theoretical
studies [
13, 14, 17] indicate that Dirac bands can b e de-
signed to occur under realistic conditions. For example,
elementary tight-binding calculat i ons indicate that Dirac
cones extending for 1 meV can be obtained by tuning
the quantum dot spacing to 20 nm, a value reachable by
state-of-the-art top-down nanofabrication methods.
A completely different route to realize AG with solid-
state materials has been rece ntly followed in Ref. [
19]
(Fig.
2). These authors succeeded in maki ng AG struc-
tures with a lattice constant of a few nm by placing
CO molecules on top of a Cu substrate with the aid of
the tip of a scanning tunneling microscope (STM) [
19]
(Fig.
2a,b). By probing the density of states of the con-
fined electrons via ST M measurements (Fig.
2d) and their
evolution as a function of an applied pseudo-magnetic
field they proved t h e formation of Dirac bands with
the characteristi c Landau levels of Dirac fermions [
5, 9].
While the scr e e nin g exerted by the bulk states under-
neath the 2DEG on the Cu( 111) surface makes these
“molecular graphene” structures not ideal candidat es for
exploring many-body effects, the large versatility in the
atomic design allows unprec ed ented local control to em-
bed, map, and tune the symmetries underlying the 2D
Dirac equation. In this system, the authors estimate
U/t ∼ 0.5 using known material parameters [
19, 37].
Within the Cu syst em , there is room to increase the
strength of effe ct i ve interactions by reducing t with larger
lattices, or else t h e atomic manipulation scheme may be
extended to other substrates with lower screening effects
in a q ue st t o real iz e other interacting phases (see below).
The band structure of molec ul ar graphene can be un-
derstood by assuming that t h e superlattice potential cre-
ated by the CO molecules acts as a weak perturbation on
the parabolic band that describes the Cu surface state.
The superlattice potential is most effective at changing
the parabolic disp er si on at the edges of the superlattice
BZ. Results of such a perturbative calculation [
13] are
shown in Fig.
2c. The superlattice potential hybridizes
the six unperturbed Cu surface states which lie at the
corners of t he new BZ. The effective Hamiltonian at each
corner of the BZ is given by a 3 × 3 matrix, which gives
rise to a doublet (blue and green bands in Fig.
2c) and
a singlet (red band in Fig.
2c). The doubly degenerate
state leads to an effective Di rac equation. This “nearly
free” electron scheme can be generalized to include the
effect of strain [
19] and spin-orbit coupling [38]. Similar
approaches can be applied to describe the appearance
of D i r ac bands in AG in semiconductors [
13, 14] . The
general symmetries of the triangul ar lattice uniquely de-
termine these couplings, which have the same form as
graphene [
39, 40]. In particular, spatially patterning the
hopping via STM atom manipulation allows the genera-
tion of both gauge (pseudo) electric and magnetic fields
in molecu l ar graphene (see Fig.
2e,f). Global changes in
the lattice constant in molecular graphene add a simple
scalar potential to the Dirac Hamiltonian equivalent to
an electrical fie l d which changes the chemical potential
or “doping” [
19]. Local changes to the lattice constant
engineer a strain introducing a vector pote ntial equiva-
lent to a large perpen d ic ul ar magnetic field [
19, 41, 42]
here tu n abl e up to 60 Tesla (Fig.
2f). Finally, creating an
alternating bond structure in the form of a Kekul´e dis t or-
tion was shown (Fig. 2e) to attach mass to the former ly
massless Dirac fermions, akin to the Higgs field [
43–45].
For future experiments, the availability of semicon-
ductors (InAs, InSb, etc.) and metals (Ag, Au, etc.)
with large spin-orbit coupling creates concrete and excit-
ing possibilities to explore topological phases of artificial
matter with these approaches [14, 38, 46, 47].
Confining photons
Photonic crystals, an optical analogue of ordinary crys-
tals, offer an additional route to design energy disper-
sion relations with characteristic Dirac points [
48]. In
such crystals, the unusual transmission p roperties near a
Dirac poi nt [
23–25] were predicted and observed experi-
mentally.
The state of the art of photonic c ry st al s operat in g
in the microwave frequency range is well descri bed in
Ref. [
49]. In this work the crystal is 2D and composed of
rows of metallic cylinders, which are arranged to form
a triangular lattice. Electromagnetic waves propagat-
ing in such a periodic struct ur e, composed of metallic
cylinders with radius R = 0.25 a, where a is the lattice
constant, exhibit a dispersion relation with several Dirac
points. In the vicinity of a Dirac point, the measured
reflection spectra resemble the STM spectra of graphene
flakes [
50, 51]. In a subsequent work [52] extremal trans-
mission through a microwave photonic crystal and the
observation of edge states close to Dirac points were also
demonstrated. The authors of Ref. [
52] have shown that
the transmission through this crystal displays a pseudo-
diffusive [
24] 1/L dependence on the thickness L of the
crystal. In addition, they measured the eigenmode in-
tensity distributions in a rectangular microwave billiard
that contains a triangular photonic cr y st al . Close to the
Dirac point there appear states at t he straight edge of the
photonic crystal that represent the artificial counterpart
of the states at a zigzag edge of natural graphene. Op-
tical analogues of graphene operating in the microwave
frequency range have been re cently used to simulate
anisotropic honeycomb lattices and to observe topologi-
cal phase transitions of Dirac points [
53].
Since 2007 intensive studies of honeycomb lattices con-
4
FIG. 2: Designer Dirac fermions in molecular graphene. (a) Artifi c ia l “molecular” graphene [19] is fabricated via
atom manipulation, and then imaged and locally probed via scanning tunneling microscopy (STM). Carbon monoxide (CO)
molecules are ind iv id u a l ly positioned (blue arrow) with the STM tip into a triangular lattice on a Copper substra te . (b) STM
topograph of molecular graphene during assembly, showing 2D surface-state electron s repelled from the molecules a n d guided
into a honeycomb lattice (black lines). (c) Band structure of molecular graphene cal c u la t ed using the nearly-free electron (NFE)
model applied to the periodically-perturbed surface electron gas [
13]. The relevant parameters have been chosen to match the
experimental results in Ref. [
19]. A Dirac band crossing appears at the K point of the superlattice Brillouin zone. Similar
band structures can also be obta in ed in the tight-binding limit [
13, 14] and ma p ped to the NFE model [19]. Progressively more
exotic variants of graphene have been fabricated using this method [
19]: (d) pristine quasi-neu tra l graphene exhibits eme rgent
massless Dirac fermions (d = 1 9 . 2
˚
A, t = 90 meV, t
′
= 16 meV); (e) graphene with a Kekul´e distortion (t
1
= 2t
2
) dresses the
Dirac fermions with a scalar gauge field creating mass (0.1±0.02 m
e
, m
e
being the bare elec tro n mass in vacuum); (f) graphene
with a triaxial strain distortion embeds a vector gauge field condensing a time-reversal-invariant relativistic quantum Hall
phase (shown here for a large pseudo-magnetic field of 60 Tesla). In the the o ry panel, images are color representations of the
strength o f the effective Carbon-Carbon bonds (corresponding to tight - b in d i n g hopping parameters t), and the curves shown
are calculated electronic density of states (DOS) from tight-binding (TB) theory. Insets show gapless and g ap ped Dirac cones
matching the e xperimental data. In the experiment panel, images are STM topogra p h s acqu i red after molecu la r assembly
(100
˚
A field of view, T = 4.2 K), and the curves are n o rm a li ze d tunnel conductance spectra ob t a in ed from the associated
nanomaterial.
structed by the method of optical induction have been
performed. The honeycomb structure in Ref. [
26] was
induced by the intensity pattern I(x, y) of three interfer-
ing pl an e waves, which is translated into a change in the
refractive index ∆n through the nonl in ear i ty in a pho-
torefractive crystal. Such a lattice exhibits several Dirac
points, which have been termed in Ref. [
26] diabolical
points after M.V. Berry and M. Wilkinson. The paraxial
evolution of the complex amplitude Ψ of a prob e beam
propagating in the lattice is governed by a normalized
Schr¨odinger-type equation of the form:
i
∂Ψ
∂z
+ ∇
2
⊥
−
V
0
Ψ
1 + I(x, y) + |Φ|
2
= 0 , (1)
where V
0
controls the relative value s of potential depth
and nonlinearity strength. The wave dynamics i n such
honeycomb photonic lattices has been e x te ns i vely stud-
ied [
26] offering evidence of the unique phenomenon of
conical diffracti on around the singular diabolical (Dirac)
points connecting the firs t and second bands. These find-
ings have represented the first experiment al observation
of the phenomenon of conical di ffr act i on, predicted by
5
strain
..."
..."
eigenvalue number
L1
L2
L-1
L-2
L0
-0.7
0.7
0
-0.7!
0.7!
0!
eigenvalue number
0 300 600
0 300 600
(a) (b)
(c) (d)
β
c
0
β
c
0
β
c
0
0
ω
2
ω
−
ω
− 2
ω
k
x
a
k
y
a
FIG. 3: Landau levels of photons in strained photonic graphene. (a) Dirac cone in spatial spectrum of unstra i n ed
photonic gra p h en e, as obtained by solving Eq. (
1) with the Ansatz Ψ(x, y, z) = φ(x, y) exp(βz). As in Fig. 1, c
0
plays the role
of the hopping parameter t. (b) Spectrum of inhomogeneously strained photonic grap h en e , accordin g to a strain tensor that
gives a constant pseudo-magnetic field. The planes are highly degenerate photonic Landau levels [
58]; (c) Eigenvalues in the
unstrained lattice are listed in descending order. Insets show a microscope image of the photonic lattice (top right) and strong
diffraction of light through the lattice (bottom left); (d) E ig envalues in th e strained lattice listed in de sc en d in g order, showing
clear evidence of Landau levels (labeled by “Ln” with n = −2, −1, 0, +1, +2, etc). Microsc o pe image of the strained lattice is
shown (inset top right), as well as resulting experimental output showing strong optical confinement in Landau level band gaps
(bottom left). Courtesy of M. Segev a n d M. Rechtsman.
W.R. Hamilton in the 19th century, arising here solely
from a periodic potential. In addition, “honeycomb gap
solitons”, residing in the gap between the second and the
third band, were observed, reflecting yet another special
property of honeycomb photonic lattices.
In recent years several t he ore t ic al papers were pub-
lished concerning various aspects of the physics of hon-
eycomb photonic lattices. These include studies of i) P T -
symmetry (i.e. symmetry under the combined operations
of parity and time reversal), ii) nonlinear wave dynam-
ics, iii ) the persistence of the Kl ei n effect, and iv) the
breakdown of conical diffraction due to nonlinear inter-
actions [
54–56]. Perhaps the culmination of these studies
has been presented in Refs. [
57–59]. Ref. [57] describes
combined theoretical an d experiment al work on the cre-
ation and destruction of topological ed ge states in “op-
tical graphene”, where, af t er the application of uniaxial
strain (compression), two Dirac points merge resulting
in the formation of a band gap. Effectively, edge states
are created (destroyed) on the zig-zag (“bearded”) edge
of the structur e. Moreover, the authors of Ref. [
57] have
claimed the observati on of a novel type of “bearded” edge
state, which cann ot be explained by th e standard tight-
binding the ory, while they can be classified as Tamm
states lacking any surface effect. This is an example that
highlights how AG structures might provide insights on
physics beyond that displayed by natural graphene. A
second complementary work [
58] dem ons tr at es the cre-
ation of synthetic magnetic fields and “photonic Lan-
dau levels” separated by bandgaps in the spatial spec-
trum of the structured diel e c t ri c lattice, as illustrated
in Fig.
3. This is the photonic analog of the relativis-
tic electron Landau levels observed in strained molec-
ular graphene [
19] (Fig. 2f). Finally, a photonic Flo-
quet TI based on an AG structure of helical waveguides,
evanescently coupled to one an ot he r, has been proposed
in Ref . [
59]. 2D photonic TIs based on optical spin-orbit
coupling—achieved through the employment of a met a-
material composed of split-ring resonators—have been
proposed in Ref. [
60].
Confining atoms and ions
Ultracold atoms and ions represent particul ar l y excel-
lent arenas for the field of quantum simulation. Effi-
cient quantum simulators of various systems (like Bose-
or Fermi-Hubbard models, lattice spin models, etc.) have
either been realized or are within reach i n the time span
of a few years [
61].
In recent years, a great deal of attention was devoted
to simulations involving honeycomb optical lattices. The