Department of physics, National Cheng Kung University, Taiwan (國立成功大學 物理系)
Tay-Rong Chang (張泰榕)
Topological crystalline insulator:
from symmetry indicators to material discovery
2018/Dec./6
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Outline
1. Topology in condensed matter physics
Basic concepts and properties of topological band structure Example: Bi2Se3
2. Topological crystalline insulator
Recent prediction: Ca2As family
3. Type-II nodal line semimetal
Recent prediction: Mg3Bi2
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slides number
Topology in condensed matter physics
Math => real space
Gauss-Bonnet Theorem:genus g = 0 g = 1 g = 1
Phys => momentum space
A B
B A
(1) (2) (3)
B A
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物理雙月刊 藏在邊緣的物理:拓樸材料與拓樸能帶理論
http://pb.ps-taiwan.org/catalog/ins.php?index_m1_id=5&index_id=235
Topology in condensed matter physics
A-orbital B-orbital (1)
Brillouin zone
wavefunction is smoothly
Brillouin zone (2)
wavefunction is NOT smoothly A-orbital B-orbital
band inversion
band inversion (Se-orbital)
(Bi-orbital)
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Obey chemical bond order
Violate chemical bond order
Topology in condensed matter physics
Gauss-Bonnet Theorem:
TKNN theory: Topological invariant number
Math => real space Phys => momentum space
Berry curvature
wavefunction is smoothly : n = 0 =>
wavefunction is NOT smoothly : n = integer =>
Topological trivial (normal insulator)
Topological nontrivial (topological insulator)
Bloch wavefun
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Γ 𝑋
𝑋 𝑀 𝑅
𝑈 𝑈
𝑍
+Γ(+) +
−−
−X(+)
− ++
−X(+)
− ++
−M(+)
− ++
+Z(+) +
−−
−U(+)
− ++
−U(+)
− ++
−R(+)
− ++
EF
Γ 𝑋
𝑋 𝑀 𝑅
𝑈 𝑈
𝑍
+Γ(−)
− +−
−X(+)
− ++
−X(+)
− ++
−M(+)
− ++
+Z(+) +
−−
−U(+)
− ++
−U(+)
− ++
−R(+)
− ++
EF Band inversion at Γ
Normal insulator (Z2=0)
Topological insulator (Z2=1)
L. Fu, PRB 76, 045302 (2007)
Calculating invariant number
𝛿Γ𝛿𝑋𝛿𝑀𝛿𝑌𝛿𝑍𝛿𝑈𝛿𝑇𝛿𝑅 = (−1)𝑍2
Topology in condensed matter physics (bulk-edge correspondence)
Surface/Interface
The gapless surface state is the hallmark of topological phase.
p
s s
p
gapless surface state
p s p
s
inversion point (gapless point) electron move
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)
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Z2=0 Z2=1
Topological insulator: Bi 2 Se 3
ARPES Theory
Surface: gapless surface states spin-momentum locked Bulk: insulating gap
topological Z2 invariant
odd/even number surface states
CB SS
CB SS
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CB band
inversion
Y. Xia et al. Nature Physics 5, 398 (2009) D. Hsieh et al. Nature 460, 1101 (2009)
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Topological crystalline insulator
Γ 𝑋
𝑋 𝑀 𝑅
𝑈 𝑈
𝑍
+Γ(+) +
−−
−X(+)
− ++
−X(+)
− ++
−M(+)
− ++
+Z(+) +
−−
−U(+)
− ++
−U(+)
− ++
−R(+)
− ++
EF
Γ 𝑋
𝑋 𝑀 𝑅
𝑈 𝑈
𝑍
+Γ(+) +
−−
−X(−) +
−+
−X(−) +
−+
−M(+)
− ++
+Z(+) +
−−
−U(+) +
−+
−U(+) +
−+
−R(+)
− ++
EF Band inversion at 𝑋and 𝑈
Normal insulator (Z2=0)
??? (Z2=0)
L. Fu, PRB 76, 045302 (2007)
L. Fu, PRL 106, 106802 (2011)
Normal band insulator?
𝛿Γ𝛿𝑋𝛿𝑀𝛿𝑌𝛿𝑍𝛿𝑈𝛿𝑇𝛿𝑅 = (−1)𝑍2
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Topological crystalline insulator
Band inversion
odd
even
Topological insulator
Additional Symmetry?
no
yes
Normal band insulator
Topological crystalline
Insulator (TCI) (Z2=1)
(Z2=0)
(Z2=0)
(𝑛 ≠ 0)
1. Mirror Chern number 2. Rotational Chern number
…etc (ignore weak TI here)
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SnTe: mirror symmetry
T. H. Hsieh et al., Nature Commun. 3, 1192 (2012) S.-Y. Xu et al., Nature Commun. 3, 982 (2012) SnTe
+Γ(+) +
−−
−𝐿(−) +
−+
−L(−) +
−+
−X(+)
− ++
+X(+) +
−−
−L(+) +
−+
−L(+) +
−+
−Z(+)
− ++
EF Z2=0
Mirror Chern number
where 𝑛∓𝑖 = mirror eigenvalue
Even number of times band inversion + additional crystalline symmetry
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TCI beyond (glide) mirror symmetry ?
TCI: mirror and glide mirror
T. H. Hsieh et al., Nature Commun. 3, 1192 (2012) S.-Y. Xu et al., Nature Commun. 3, 982 (2012)
SnTe => mirror KHgX(X=As,Sb,Bi) => glide mirror
Z. Wang et al., Nature 532, 189 (2016)
J. Ma et al., Science Advances 3, e1602415 (2017)
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New TCI: protected by the N-fold rotational symmetries Chen Fang and Liang Fu, arXiv:1709.01929 (2017).
TCI: rotational symmetry
C2 C4 C6
Real material ???
Topological invariant number is not convenient to calculate from DFT.
Cn = n Dirac points on one surface
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TCI: symmetry indicator
(symmetry indicator)
How to implement on first-principle?
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Zhida Song, Tiantian Zhang, Zhong Fang, Chen Fang:
Nature Communications 9, 3530 (2018)
Systematic method (Fu-Kane-like formula) for searching TCI based on symmetry indicator
TCI: symmetry indicator
when certain additional symmetry Y is present, topological invariants of TCIs protected by symmetry X can be inferred by the Y -symmetry eigenvalues of energy band.
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I4/mmm (#139) Ca2As: body-centered tetragonal lattice
Time-reversal Inversion
M(100) M(010) M(001) M(110) M(1-10)
C4(001) C2(100) C2(010) C2(001) C2(110) C2(1-10) Symmetry
operation
TCI: candidate material Ca 2 As
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Energy (eV)
Counting the irred. rep. number of each band
The symmetry indicator of # 139 is (Z2,Z8) Step-1
Ca 2 As: symmetry indicator
(Z2,Z8)=(0,4) for Ca2As
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Step-2 mirror glide mirror rotation
Ca 2 As: symmetry indicator
I4/mmm (#139)
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Step-3
Ca 2 As: symmetry indicator
I4/mmm (#139)
mirror Chern number
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Topological surface states
I4/mmm (#139)
C4 + Mirror
Energy (eV)
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Topological surface states
I4/mmm (#139)
C4 + Mirror
C2 + Mirror
Energy (eV)
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Breaking mirror & C4
Breaking mirror
TCI: rotational symmetry protected
C2+Mirror C2
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Ca2As Ca2Bi
Sr2Sb
Phase transition
Topological phase transition
TCI
TCI+Weak TI Normal insulator
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(1) Searching TCI by using symmetry indicator for the first time Ca2As (Z2,Z8)=(0,4)
(2) Topological surface states are protected by multi topological invariant numbers
(3) It is first time that the Topological surface states are protected by rotational symmetry
Breaking mirror
Conclusion (TCI)
arXiv:1805.05215 (2018)
X. Zhou et al., Phys. Rev. B 98, 241104(R) (2018)
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Nodal-line semimetal (節線半金屬)
Normal metal: 2D Fermi surface Topological metal: 1D Nodal-line
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NL
PbTaSe 2 TlTaSe 2
Nodal-line semimetal
G. Bian et al., Nature Commun. 7, 10556 (2016) G. Bian et al., Phys. Rev. B 93, 12113(R) (2016)
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Type-I vs Type-II
Type-II DP
Slope sign: opposite Slope sign:
the same
Type-II
Type-I
Slope sign: opposite Slope sign: opposite
Type-II Weyl: A. A. Soluyanov et al.,Nature volume 527, 495 (2016) Type-II Dirac: T.-R. Chang et al., Phys. Rev. Lett. 119, 026404 (2017)
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n-doping
Type-II Nodal-line semimetal
Mg 3 Bi 2
T.-R. Chang et al., Advanced Science (2018), DOI: 10.1002/advs.201800897
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Dr. Xiaoting Zhou (NCKU)
Chuang-Han Hsu (NUS)
Prof. Hsin Lin (Academia Sinica)
Dr. Su-Yang Xu (MIT)
Prof. Liang Fu (MIT)
MOST(Taiwan) Young Scholar Fellowship:
MOST Grant for the Columbus Program
Acknowledgements (TCI)
X. Zhou et al., Phys. Rev. B 98, 241104(R) (2018)
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Acknowledgements (Nodal-line)
MOST(Taiwan) Young Scholar Fellowship:
MOST Grant for the Columbus Program Prof. R. J. Cava
(Princeton U.)
Prof. Weiwei Xie (Louisiana State U.)
Prof. Guang Bian
(University of Missouri)
T.-R. Chang et al., Advanced Science (2018), DOI: 10.1002/advs.201800897
Thank you !
To see a world in a grain of sand … -William Blake
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