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Nanostructured Carbon Allotropes as
Weyl-Like Semimetals
Shengbai Zhang
Department of Physics, Applied Physics & Astronomy
Rensselaer Polytechnic Institute
symmetry
In quantum mechanics, symmetry can be critically important.
Consider, for example, a two-particle system. Under particle
exchange, bosons satisfy  1 , 2 =  2 , 1 , whereas
fermions satisfy  1 , 2 = − 2 , 1 .
Just because the symmetry, bosons and fermions are very
different from classical particles.
Recent development in solid-state physics is also about
symmetry: in case of Weyl fermions, chiral symmetry leads to
novel properties/topological protection not envisioned before.
2
topological protection
a normal ring
 One cannot deform
a normal ring into a
Mobius ring without
cutting the strip and
a Mobius ring
then rejoin.
3
solid and its bloch wave
A solid is a periodic system in which electron
wavefunction may be written as a Bloch
wave   =  ∙   =  ∙   .
The reciprocal space of  is the  space in
which the smallest repeating unit defines
the Brillouin zone.
Brillouin zone (BZ) forms a closed loop in the
sense that a -point exiting from one face of the BZ is equivalent
to reentering from the opposite face of the BZ.
4
berry curvature & chiral quantum number
Electron living inside the Brillouin zone feels an (effective)
vector potential   =      
(Berry connection).
The corresponding (Berry) curvature   =  ×  
defines a field (similar to a magnetic field ).
Chiral quantum number is defined by χ =
1

2 
 ∙  
(Chern number).
Non-trivial Chern number (+1 or −1) tells the chirality (righthanded or left-handed).
5
low-energy particles
Tomorrow's condensed matter physics will be rooted in many-
body physics – studying collective excitations and quasiparticles
Theory developed for elementary particles can be shipped here
to explore many-body interactions which cannot be easily
obtained from experiment via serendipity. → Theory is and will
lead the way.
Compared to traditional particle physics, condensed matter is a
much richer field and experimental test should be easier. → An
emerging battlefield of particle physics.
Good condensed matter physicist needs to be intradisplinary!
6
spin-half dirac fermions
Dirac Hamiltonian (3D):
 2
=
 ∙ 
 ∙ 
=
− 2
 2
0
0
 2

  − 
  + 
−
Massless Dirac fermions ( = 0),  =
0
 ∙ 

  − 
  + 
−
− 2
0
0
− 2
 ∙ 
0
Dirac semimetal (4x4 matrix; doublydegenerate Dirac cones; linear
(2-Dirac points)
dispersion in all three directions).
M. Koshino
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3-dimensional spin-half weyl fermions
NJP 9, 356 (2007)
Separate in kspace the two
Dirac cones :
Now, each cone is described by a 2x2 Weyl Hamiltonian: ± =

±
 + 
 − 
. Weyl fermions are protected by the chiral
−
symmetry discussed earlier, so they are robust against perturbations.
8
surface states: fermi arcs
 Weyl points of different chirality's can be viewed as magnetic
monopoles (MMP) with ±charges
 On the surfaces of a slab, Fermi arcs appear, which is characteristic
of Weyl semimetal.
y
A state moving in the (+y)-direction on
top surface cannot be scattered back.
 To observe Weyl points, the
material must do not have either
time reversal symmetry or
inversion symmetry.
9
why bother?
 Weyl semimetal exhibits chiral anomaly, meaning chiral “charges” are not
conserved. One can use  ∥  fields to pump charge from one chiral
channel to another.
 Experimental ramifications include negative magnetoresistance;
quantum anomalous Hall effect (for dissipationless carrier transport like
superconductivity); non-local transport; and non-conservation of
ordinary current (at least locally).
 It can also give rise to
unusual optical conductivity,
and many more.
real-part of
optical
conductivity 
versus chemical
potential  &
optical
frequency .
10
dirac/weyl semimetal as an intermediate phase
 A Dirac or Weyl
semimetal is an
intermediate phase in
the transition between a
topological insulator (TI)
phase and a normal
insulator (NI) phase.
 Topological insulator are materials with large spin-orbit coupling (SOC),
such as Bi2Te3
 Search for Weyl semimetals also follows the same line of thought.
11
in pursuit of weyl semimetal (current status )
Photonic crystal: Science 349, 622 (2015)
30-meV gap
TaAs (Ta is heavy, large spin-orbit coupling):
Science 349, 613 (2015); PRX 5, 031013 (2015).
TaAs Band Structure (E versus k)
(eV)
Drawbacks with TaAs:
 Too many Weyl points (24 in total) and they are too close (only ~% of BZ),
unlike graphene
 Trivial Fermi pockets at Fermi level dominate (zero gap in the DOS).
12
what might be the next?
discovery of Weyl semimetal is
an APS Highlight of the Year
 You bet, the study of Weyl
fermion will continue
# papers with “Weyl” in topic
 Regardless, experimental
# papers with “graphene” in topic:
1945-2016
0 (1945-1990);
95,366 (1991-2016);
1,803 (2016 alone);
>100 per day now.
13
graphene: an example of 2d dirac
weyl semimetal
-space: A,B sublattice
-space:  ≠ ′
Near  and ’:
0
=
 + 
 − 
−
0
2 Dirac cones separated by
1
3
 −  .
K’
K
14
carbon allotropes
Graphene
Graphite
+ many more …
Carbon has
many allotropes
due to its
exceptionally
strong C-C
bonds. Once
formed, these
allotropes are
hard to break.
Carbon nanotube
C60
15
dp in graphene: unique orbital interactions
 
graphene →
atomic  -orbital
→ 2D Dirac point
atomic  orbitals


graphene network
→ atomic ( ,  )orbitals → 3D
Weyl point?
16
so, here is our charge
 Search for systems whose orbital interactions have the
form of 3D Weyl Hamiltonian
 However, different from others searching for existing
materials of limited use and supply, we target materials
of broad use but in forms not yet synthesized
 In the process, we also uncovered structures with
Hamiltonians that do not fit into any of the currently
known models.
17
why weyl-like semimetal?
 Light elements have exceedingly small spin-orbit
coupling (SOC for carbon ~1 millikelvin)
 At room temperature, (thermal scattering)/SOC is
300/0.001 = 3 × 105
 Hence, for most applications, spin may be treated as a
dummy variable, leading to Weyl-like semimetals
 Conclusion: Weyl physics exists in solids made of purely
light elements (the lighter, the better).
18
interpenetrated graphene network (IGN)
3
2
Formation
energy (eV/C)
Graphene
Diamond
IGN
C60
0.0
0.13
0.23
0.37
 No imaginary-frequency phonons → kinetically stable.
19
weyl line nodes
Brillouin zone

Z
T
G
Y
Weyl wedge

 Line nodes are the only Fermi surface
for the entire system, along which the
energy dispersion looks like a wedge.
Nano Letters 15, 6974 (2015)
20
emergence of weyl points
 One may break the inversion symmetry either (1) by
displacing some carbon atoms or (2) by inserting He
interstitials, both turn the Weyl line nodes into points
(1)
(2)
21
fermi arcs appear on the [010] surface
Schematic
Nano Letters 15, 6974 (2015)
Actual calculation
22
back to line nodes: what is unique?
Chern number: χ =
1
2
  ∙  
Co-dimension = 2 (even) → weaker but non-vanishing topological
protection.
Line node gives rise to topologicallyprotected flat bands on surfaces. On
PRB 82, 184502 (2010)
a flat band, Coulomb repulsion U is
exceptionally large, leading to a
strong electron-electron correlation
Can you imaging strongly-correlated carbon?
23
line nodes are robust under uniaxial strain

→ reducing 
 Topological protection
remains until a critical
angle  = 64°.
24
at  = 64° transition from IGN to CKL
Interpenetrated graphene
network (IGN)
1/3 C 4fold; 2/3 C 3fold (sp2)
Nano Letters 15, 6974 (2015)
Carbon Kagome lattice (CKL)
G-3
All C 4fold (sp3)
PRL113, 085501 (2014)
25
understanding electronic properties
 Polyacetylene is the root for most carbon allotropes. By connecting
the chains, one
builds graphene,

diamond, IGN,
anti-bonding state
Kagome lattice, etc.
G
 Different
A(=Z)
characters of the
occupied and
empty states are
the reason for
bonding state
Weyl semimetal.
26
band inversion yields topological protection
Increasing uniaxial strain
1D→3D
IGN
CKL
polyacetylene

A
G
Red =  1
Blue =
2
Green =  3
27
ckl is as remarkable as ign
 3.4-eV direct gap
(blue color)
(by HSE calc.)
 Effective mass
∗ ’s comparable
to Si
 Light absorption
comparable to
GaN
 Realization of optical-electronic integration in one material.
Phys. Rev. Lett. 113, 085501 (2014)
28
frustration in triangular lattice
-orbital
carbon Kagome lattice
Frustration in life
+
−
-down
-up
Frustration in physics
 Upon doping, will carriers in CKL act as spin-like liquid?
29
there is a whole family of 3d-graphene weyl semimetals
 2-type carbon
rings (more
stable than IGN)
 Weyl surface
nodes as Fermi
level
 Co-dimension =
1 (odd) → more
topological
protection (?)
Colors denote different
types of carbon
30
ordered
random
(b)
(a)
Transmission
Electron
Microscopy (TEM)
images of carbon
channels. Most of
the channels are
perpendicular to
the page.
Featured in Physics
& Editor’s Suggestion
31
summary
Weyl-line nodes in
interpenetrated carbon
network [Nano Letters 15,
6974 (2015)]
Carbon Kagome lattice
[Phys Rev Lett 113, 085501 (2014)]
Weyl-surface nodes in three-dimensional
graphene [under review by Nanoscale (2016)]
32
acknowledgements
Vincent Meunier
Marvin Cohen, UCB
Yuanping Chen
Yiyang Sun
Han Wang
Damien West
Rensselaer Polytechnic Institute
S.Y. Yang,
Singapore
Fan Zhang,
UT Dallas
33