We outline an approach to endow a plain vanilla material with topological
properties by creating topological bands in stacks of manifestly nontopological
atomically thin materials. The approach is illustrated with a model system
comprised of graphene stacked atop hexagonal-boron-nitride. In this case, the
Berry curvature of the electron Bloch bands is highly sensitive to the stacking
configuration. As a result, electron topology can be controlled by crystal axes
alignment, granting a practical route to designer topological materials. Berry
curvature manifests itself in transport via the valley Hall effect and
long-range chargeless valley currents. The non-local electrical response
mediated by such currents provides diagnostics for band topology

Levitov, Leonid S.

Samutpraphoot, Polnop

Song, Justin C. W.

English

arXiv

We outline a designer approach to endow widely available plain materials with topological properties by stacking them atop other nontopological materials. The approach is illustrated with a model system comprising graphene stacked atop hexagonal boron nitride. In this case, the Berry curvature of the electron Bloch bands is highly sensitive to the stacking configuration. As a result, electron topology can be controlled by crystal axes alignment, granting a practical route to designer topological materials. Berry curvature manifests itself in transport via the valley Hall effect and long-range chargeless valley currents. The nonlocal electrical response mediated by such currents provides diagnostics for band topology

Caltech Authors - Main

Supporting Information
Song et al. 10.1073/pnas.1424760112
Computing Berry Curvature
The topological properties of G/h-BN can be analyzed through
the Berry curvature in the minibands. Even though the G/h-BN
Hamiltonian, Eq. 2 of the main text, possesses TR symmetry, its
broken inversion symmetry allows for a finite Berry curvature to
develop in the SBZ
ΩnðkÞ=∇k ×AnðkÞ, AnðkÞ= ihunðkÞj∇kjunðkÞi. [S1]
Here n is the band index, A is the Berry connection, and junðkÞi
are the eigenvectors of Eq. 2 of the main text. In the main text,
we concentrate on a single valley and the lowest conduction
miniband (labeled “1” in Fig. 2B of the main text).
Here we comment on the procedure used to evaluate the Berry
curvature. We use the eigenvectors, junðkÞi, obtained at each k
from numerical diagonalization of the Hamiltonian, Eq. 2 of the
main text [in doing so we use ð2m+ 1Þ2 points in the extended
superlattice Brillouin zone scheme, typically with m= 3]. Next,
we calculate ΩnðkÞ using a gauge-invariant method similar to that
used in ref. 28. We summarize this method briefly: Adopting a fine
mesh of the Brillouin zone (e.g., in Fig. 3 of the main text the mesh
for the grid shown was 400× 460), the Berry curvature at each k
can be obtained by numerically integrating the Berry connection
AðkÞ in small loops around k (viz., Stokes’ theorem). For each
loop, we first choose a set of eigenvectors around the loop and then
calculate Berry’s connection between points on the loop. This
ensures that the arbitrary phases appear twice in the loop integral
but with opposite sign. In this way (numerical) problems with gauge
choice are eliminated as arbitrary phases introduced through the
numerical diagonalization are canceled in the loop.
k ·p Theory for SBZ ~K, ~K′ Points
The finite Berry curvature at ~K, ~K′ (in the SBZ) can be traced to
mixing of the pseudospin textures by Bragg scattering shown in Fig.
S1A. The bandstructure near ~K, ~K′ can be modeled using the k · p
method. Both ~K and ~K′ are points of triple band crossing in the
limit of vanishing superlattice potential. We therefore introduce
three states jji, j= 1,2,3, one for each crossing band. Using band
dispersion linearized near ~K (or, ~K′) and introducing a matrix ele-
ment describing the coupling of the three states by Bragg scattering,
we can write the k · p Hamiltonian in the conduction band as
H =
X
j=1,2,3
EjðpÞj jij ji+
X
j′−j=1ðmod3Þ
uj j′j ji+ h.c., [S2]
where EjðpÞ= hψðejÞjvσ · pjψðejÞi= vp · ej, and u= uj, j′ = hψðejÞ
jmσ3jψðej′Þi= ið
ffiffiffi
3
p
=4Þm3. In the latter, j≠ j′ and are taken cycli-
cally. Here jψðejÞi is a spinor pointing along ej and e1, e2, e3 are
three unit vectors pictured in Fig. S1A. Evaluating the determi-
nant in
det
0
B@
E1ðpÞ− e u u*
u* E2ðpÞ− e u
u u* E3ðpÞ− e
1
CA= 0, [S3]
gives
ðE1ðpÞ− eÞðE2ðpÞ− eÞðE3ðpÞ− eÞ+ u3 + u*3 + 3juj2e= 0. [S4]
The resulting bandstructure is shown in Fig. S1B, which
mimics the behavior close to the ~K , ~K ′ seen in Fig. 2B of the
main text as expected.
We use eigenstates obtained from the above Hamiltonian, Eq.
S2, to compute Berry curvature as shown in Fig. S1C. Here we
adopted the same numerical method as outlined in the above
section. The Berry curvature distribution shown in Fig. S1C is
concentrated close to the avoided crossings as expected from
Fig. 3 of the main text.
Summing up the Berry curvature in each of the bands, we
directly verify that bands 1 and 3 contribute −π=2 and π=2 to the
net flux
R
d2qΩðqÞ, respectively, whereas the net flux in band 2 is
zero. In Fig. S1 we used m3 > 0. Flipping the sign of m3 results in
an opposite sign of flux in band 1 and 2 to that shown in Fig.
S1C; band 2’s net flux remains zero. As a result, we find that the
net Berry flux in band 1 (the lowest conduction band) is de-
scribed by Eq. 1 of the main text.
Importantly it is m3, which controls the Bragg scattering at
~K , ~K ′, that determines the Berry curvature at the edges of the
SBZ. As discussed in the main text, the separate origins of Berry
curvature at ~Γ and ~K , ~K ′ points allows for control over Cv of the
reconstructed minibands of G/h-BN.
Song et al. www.pnas.org/cgi/content/short/1424760112 1 of 2
Fig. S1. Bandstructure and Berry curvature around ~K, ~K′ points. (A) Bragg scattering at SBZ corner points ~K, ~K′ mixes the three neighboring SBZs in the
extended zone scheme. The avoided crossing of three bands at ~K, modeled by the k ·p Hamiltonian, Eq. S2, yields a bandstructure shown in B (energy and
wavenumber units are juj and juj=v). (C) Berry curvature distribution, ΩðqÞ, in reciprocal space features threefold symmetry for bands 1 and 3 and sixfold
symmetry for band 2. The net Berry curvature (Berry flux) is −π=2 and π=2 for bands 1 and 3, and zero for band 2.
Song et al. www.pnas.org/cgi/content/short/1424760112 2 of 2


Topological Bloch Bands in Graphene Superlattices

Significance
          A family of designer topological materials is introduced, comprising stacks of two-dimensional materials which by themselves are not topological, such as graphene. Previously, topological bands in graphene were presumed either impossible or impractical. The designer approach turns graphene into a robust platform with which a host of topological behavior can be realized and explored.</jats:p

Justin C. W. Song

Polnop Samutpraphoot

Leonid S. Levitov

Crossref

Proceedings of the National Academy of Sciences

Topological Bloch bands in graphene superlattices

We outline a designer approach to endow widely available plain materials with topological properties by stacking them atop other nontopological materials. The approach is illustrated with a model system comprising graphene stacked atop hexagonal boron nitride. In this case, the Berry curvature of the electron Bloch bands is highly sensitive to the stacking configuration. As a result, electron topology can be controlled by crystal axes alignment, granting a practical route to designer topological materials. Berry curvature manifests itself in transport via the valley Hall effect and long-range chargeless valley currents. The nonlocal electrical response mediated by such currents provides diagnostics for band topology.National Science Foundation (U.S.). Science and Technology Center for Integrated Quantum MaterialsNational Science Foundation (U.S.) (Grant DMR-1231319)Massachusetts Institute of Technology. Institute for Soldier Nanotechnologies (Contract W911NF-13-D-0001

Levitov, Leonid

DSpace@MIT

https://authors.library.caltech.edu/58759/3/pnas.201424760SI.pdf

Topological Bloch Bands in Graphene Superlattices

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