Download Nanostructured Carbon Allotropes as Weyl

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 𝜓 𝑟1 , 𝑟2 = 𝜓 𝑟2 , 𝑟1 , whereas
fermions satisfy 𝜓 𝑟1 , 𝑟2 = −𝜓 𝑟2 , 𝑟1 .
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.
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solid and its bloch wave
A solid is a periodic system in which electron
wavefunction may be written as a Bloch
wave 𝜓𝑘 𝑟 = 𝑒 𝑖𝑘∙𝑟 𝑢𝑘 𝑟 = 𝑒 𝑖𝑘∙𝑟 𝑢 𝑘 .
The reciprocal space of 𝑟 is the 𝑘 space in
which the smallest repeating unit defines
the Brillouin zone.
Brillouin zone (BZ) forms a closed loop in the
sense that a 𝑘-point exiting from one face of the BZ is equivalent
to reentering from the opposite face of the BZ.
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berry curvature & chiral quantum number
Electron living inside the Brillouin zone feels an (effective)
vector potential 𝐴 𝑘 = 𝑖 𝑢 𝑘 𝛻𝑘 𝑢 𝑘
(Berry connection).
The corresponding (Berry) curvature 𝐹 𝑘 = 𝛻𝑘 × 𝐴 𝑘
defines a field (similar to a magnetic field 𝐵).
Chiral quantum number is defined by χ =
1
𝐹
2𝜋 𝐹𝑆
𝑘 ∙ 𝑑 𝑆𝑘
(Chern number).
Non-trivial Chern number (+1 or −1) tells the chirality (righthanded or left-handed).
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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!
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spin-half dirac fermions
Dirac Hamiltonian (3D):
𝑚𝑣 2
𝐻=
𝑣𝜎 ∙ 𝑝
𝑣𝜎 ∙ 𝑝
=
−𝑚𝑣 2
𝑚𝑣 2
0
0
𝑚𝑣 2
𝑣𝑝𝑧
𝑣 𝑝𝑥 − 𝑖𝑝𝑦
𝑣 𝑝𝑥 + 𝑖𝑝𝑦
−𝑣𝑝𝑧
Massless Dirac fermions (𝑚 = 0), 𝐻 =
0
𝑣𝜎 ∙ 𝑝
𝑣𝑝𝑧
𝑣 𝑝𝑥 − 𝑖𝑝𝑦
𝑣 𝑝𝑥 + 𝑖𝑝𝑦
−𝑣𝑝𝑧
−𝑚𝑣 2
0
0
−𝑚𝑣 2
𝑣𝜎 ∙ 𝑝
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: 𝐻± =
𝑝𝑧
±𝑣
𝑝𝑥 + 𝑖𝑝𝑦
𝑝𝑥 − 𝑖𝑝𝑦
. Weyl fermions are protected by the chiral
−𝑝𝑧
symmetry discussed earlier, so they are robust against perturbations.
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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.
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why bother?
 Weyl semimetal exhibits chiral anomaly, meaning chiral “charges” are not
conserved. One can use 𝐵 ∥ 𝐸 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 𝜎𝑥𝑥
versus chemical
potential 𝜇 &
optical
frequency 𝜔.
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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.
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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).
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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.
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graphene: an example of 2d dirac
weyl semimetal
𝑟-space: A,B sublattice
𝑘-space: 𝐾 ≠ 𝐾′
Near 𝐾 and 𝐾’:
𝑝0𝑧
𝐻=𝑣
𝑝𝑥 + 𝑖𝑝𝑦
𝑝𝑥 − 𝑖𝑝𝑦
−𝑝
0𝑧
2 Dirac cones separated by
1
3
𝒃𝟏 − 𝒃𝟐 .
K’
K
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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
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dp in graphene: unique orbital interactions
𝑝𝑧 𝑝𝑧
graphene →
atomic 𝑝𝑧 -orbital
→ 2D Dirac point
atomic 𝑝 orbitals
𝑝𝑦
𝑝𝑥
graphene network
→ atomic (𝑝𝑥 , 𝑝𝑦 )orbitals → 3D
Weyl point?
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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.
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why weyl-like semimetal?
 Light elements have exceedingly small spin-orbit
coupling (SOC for carbon ~1 millikelvin)
 At room temperature, (thermal scattering)/SOC is
300𝐾/0.001𝐾 = 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).
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interpenetrated graphene network (IGN)
𝑠𝑝3
𝑠𝑝2
Formation
energy (eV/C)
Graphene
Diamond
IGN
C60
0.0
0.13
0.23
0.37
 No imaginary-frequency phonons → kinetically stable.
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weyl line nodes
Brillouin zone
𝜃
Z
T
G
Y
Weyl wedge
𝐸𝐹
 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)
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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)
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fermi arcs appear on the [010] surface
Schematic
Nano Letters 15, 6974 (2015)
Actual calculation
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back to line nodes: what is unique?
Chern number: χ =
1
2𝜋
𝐴 𝑘 ∙ 𝑑 𝑙𝑘
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?
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line nodes are robust under uniaxial strain
𝜃
→ reducing 𝜃
 Topological protection
remains until a critical
angle 𝜃 = 64°.
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at 𝜃 = 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)
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understanding electronic properties
 Polyacetylene is the root for most carbon allotropes. By connecting
the chains, one
builds graphene,
𝐸𝐹
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.
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band inversion yields topological protection
Increasing uniaxial strain
1D→3D
IGN
CKL
polyacetylene
𝐸𝐹
A
G
Red =  1
Blue =
2
Green =  3
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ckl is as remarkable as ign
 3.4-eV direct gap
(blue color)
(by HSE calc.)
 Effective mass
𝑚∗ ’s comparable
to Si
 Light absorption
comparable to
GaN
 Realization of optical-electronic integration in one material.
Phys. Rev. Lett. 113, 085501 (2014)
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frustration in triangular lattice
𝑝-orbital
carbon Kagome lattice
Frustration in life
+
−
𝑝-down
𝑝-up
Frustration in physics
 Upon doping, will carriers in CKL act as spin-like liquid?
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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
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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
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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)]
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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
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