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Electronic Topology and Coupling with Phonons in Crystals: Unusual Phonon Anomalies and Thermoelectric Response
Theoretical Sciences Unit
J Nehru Centre for Advanced Scientific Research
(JNCASR), Jakkur, Bangalore 560 064 INDIA
Umesh V Waghmare
Funded by Dept of Sci and Tech, Govt of India
Talk is focused on
• Electronic Topology in Materials • Effective single electron picture (Density Functional
Theory) • Commonality between 2-D Materials like graphene & 3-D
topological insulators & semi-metals • Material-specific theory
Insulators
Mott Insulator: Electron correlations
V2O3, LaTiO3
Anderson Insulator: Disorder in a material ► localization
Highly disordered alloys
Band Insulator: even no of electrons, periodic potential
(nearly free electron model)
U t
Single electron physics
Topological Insulators
Electronic “Band” Structure
Rohlfing et al, Phys. Rev. B48 (1993)
17791-17805
Periodic structure of a crystal:
V(r+R) = V(r)
Bloch Theorem
Bloch vector: k
n is the band-index
BZ of FCC Periodicity of Bloch functions in reciprocal space
Geometric Phases of Bloch Functions in k-space
Treat Bloch Hamiltonian H(k) parameterized by “Bloch vector”
A path in k-space going from k0 to k0+G is a closed path (due to
periodicity), i.e. eigenfunctions and eigenvalues are the same
at k0 to k0+G.
** a demo on Berry Phases or Geometric Phases.
Related to polarization or electric dipole moment = e
Electronic Topology is intimately linked with
geometric properties of electronic wave functions:
Berry connection.
Berry connection: Ak
Time Reversal Symmetry
If Hamiltonian H is invariant under time reversal symmetry,
Then, is also an eigenfuncion with the same energy E.
and
This leads to the following condition for Bloch electron:
and for the Berry Connection:
Thus, geometric properties of Bloch electrons are strongly
constrained by the time reversal symmetry.
Stokes Theorem:
Fermi Surface of a metal
Anomalous Hall
Conductivity Karplus & Luttinger; Sundaram & Niu
Geometric Phases and Anomalous Hall Conductivity:
Hall effect with out magnetic field!
E J
𝑣 = −𝑒
𝑐𝐸 × Ω(𝑘)
Electronic Structure of Graphene
𝐸 =𝑝2
2𝑚
𝐸 = 𝑝2𝑐2 +𝑚2𝑐4 If m=0, E=pc=ℏ𝑐. 𝑘
Massless Dirac (Relativistic) Fermion
Dirac Cone Conduction Band
Conduction Band
𝜓2𝜋 = 𝑒𝑖𝜋𝜓0
States at the Dirac point: Magnetic Moment!
Chirality
Conservation of chirality: High mobility
Exotic Physics of Graphene
Valleys in the Electronic Structure of Graphene
K
K
K
K’
K’
K’= -K
To access this exotic physics in graphene devices: One often needs to open up a gap by breaking inversion symmetry!
Opposite Chirality at K and -K
Xiao et al, Phys Rev Lett (2008).
Topological Transport in Graphene: Valleytronics
Valleytronics
TWO Dirac cones in Graphene
Opposite Magnetic Moment
(with broken inversion symmetry)
Schaibley et al, Nature Reviews Materials 1, 1 (2016)
2-D MoS2
Electronic Structure of 2H MoS2
Valley and
Spin
Band spin-split
Due to Spin-orbit
coupling & lack of
Inversion Symmetry Xiao et al arXiv 1112.3144(2012)
Low Energy Electronic Hamiltonian of MoS2
τ: valley index=±1 Gap Spin Orbit Coupl.
Conduction band Valence Band
Coupled Spin and Valley Physics!
Valley Hall Effect, Spin Hall Effect
Circularly polarized light can be used to excite
e- carriers selectively in a given valley: Valley-tronics
Xiao et al, PRL (2012), Wang et al Nature Comm (2012)
Valley-opto-electronics: Schaibley et al, Nature Reviews Materials 1, 1 (2016)
Electron Phonon Coupling
• Stronger effects in low-D systems
• Small or vanishing band-gaps:
- Time-scales of e and phonons are comparable
- Breakdown of adiabatic approximation
Take-home message: Subtle changes in electronic structure or topology
can be detected through Raman Spectroscopy
Electron-Phonon Coupling (EPC) in Graphene
EPC results in changes in phonon frequencies
with doping:
key to use of Raman for
characterization of devices
Breakdown of Born-Oppenheimer Approximation!
Das et al, Nat Nanotech. (2008).
Electric Field Effect Tuning of EPC
in graphene:
Gate voltage changes w of G-band
Pinczuk et al, Phys Rev Lett. (2008).
Electron-Phonon Coupling (EPC) in 1H-MoS2
Chakraborty, Bera, Muthu, Bhowmick, Waghmare and Sood,
Phys Rev B 85, R 161403 (2012) Editor’s Suggestion.
Gate field: change doping level
It affects specifically A1g mode.
Theory reproduces both w and
relative changes in line-width
expt
theory
theory
expt
Raman: Characterization of FET
Electron-Phonon Coupling: Prediction of Ferroelectricity in 1T MoS2
Trimerization of Mo in √3x√3 of metallic1T
Structure E-Flippable
Electric
Dipole
Moment
┴ to plane
Verified from first-principles
Semiconductor
e- e-
e-
e-
e-
e-
e-
e- e-
Strongly
Coupled Free Carrier e- Electric Dipoles
Ferroelectric Semiconductor
Sharmila Shirodkar & UWaghmare,
Phys Rev Lett 112, 157601 (2014)
Electronic Topology in
3-D Materials
3D-analogues of graphene: Weyl & Dirac Semi-metals
𝐻 𝑘 = ℏ 𝑣𝑖𝑗𝑘𝑖𝜎𝑗
Two linearly dispersed bands that are degenerate at the Weyl Point
Ω
Chern Number: Berry Flux passing through a sphere centered at Weyl point 𝐶 = Ω. 𝑑𝑆
= sgn (det[vij])
Time Reversal (T) Weyl Point at k Another Weyl Point at –k with same C Inversion (I): Weyl Point at k Another Weyl Point at –k with –C Both T and I symmetries: Dirac Semimetal with a 4-degeneracy
Monopoles in k-space
Wan, X., et al Phys.l Rev. B 83, 205101 (2011).
Young et al, Phys Rev Lett 108, 140405 (2012)
Dirac Semi-metal: A Parent to Weyl Semi-metal
• 4 Linearly dispersed Bands crossing at a Dirac Point • Not topological (Two opposite Chern Numbers): Gap can be
opened up, using another Dirac Matrix as a perturbing potential • Break T or I symmetries: A Dirac point separates into two Weyl Points Burkov et al Phys. Rev. Lett. 107, 127205 (2011)
4-degeneracy 2-degeneracy
Weyl Semi-metal occurs at a transition from Axion to Mott Insulator Wan, X., et al Phys.l Rev. B 83, 205101 (2011).
Balents, Physics 4, 26 (2011)
Fermi Arc at Surface
Inversion of conduction and valence bands of opposite parity
• Dirac Semi-metal occurs at a transition from Band to Topological Insulator (when Inversion symmetry is preserved)
Dirac Semi-metal: A parent to BI and TI
Designing DSMs using Space Group Symmetries: 4-degeneracy at Brillouin Zone Boundary points
Young et al, Phys Rev Lett 108, 1e40e405 (2012)
Topological Insulators (TIs)
• Time Reversal T Broken, Chern Insulators
1. known since 1985 (theoretically)
2. Integer quantum Hall effect with out B!
3. Quantum Anomalous Hall (QAH) insulators
4. No known real examples [?]
• T-symmetric TIs
1. known since 2005 [Mele and Kane]
2. Z2 Topological insulators, in 2-D
3. Strong topological insulators, in 3-D
Stokes Theorem:
Fermi Surface of a metal
Anomalous Hall
Conductivity Karplus & Luttinger; Sundaram & Niu
Geometric Phases and Anomalous Hall Conductivity:
Hall effect with out magnetic field!
E J
Geometric Phases and Anomalous Hall Insulator:
Hall effect with out magnetic field!
C: Chern Number, integer
Similarity to Integer Quantum Hall Effect
Time reversal symmetry has to be broken in Chern TI’s.
Understanding in terms of geometric phases (=polarization)
Bhattacharjee and Waghmare, PRB (2005)
Polarization P gives a surface charge:
Edge states (2d) or Surface charge (3D)
Integer quantum Hall effect
Winding Number
QAH Insulators
Candidates: Magnetic insulators, with strong spin-orbit
coupling; connection with multiferroics?
Z2 Topological Insulator
Chern Number C = 1 for spin up electrons,
= -1 for spin down electrons
It obeys time reversal symmetry
total C=0
?? Z2 invariant is odd
needs spin-orbit coupling
Edge States
θ
Normal Insulator Z2Topological Insulator
Geometric Phases of Kane-Mele Model
(Ref. A Soluyanov)
Red: Spin up
Blue: Spin down
Remember,
θ is like the expectation value of x-operator
Case of many bands: Non-Abelian Geometric Phases
𝜓 → 𝑒𝑖𝛾𝜓
[𝜓] → 𝑒𝑖Γ[𝜓]
Γ: A Non-Abelian Geometric Phase Hermitian Matrix Eigenvalues of Γ give the eigenvalues of x=xn
Abelian:
Non-Abelian:
Bhattacharjee and Waghmare, Phys. Rev. B 71, 045106 (2005)
Monitor eigenvalues xn as function of k, to determine topological invariants of electronic structure
Soluyanov et al., Phys. Rev. B, 83, 235401 (2011)
Topological Crystalline Insulators: Topological Invariants
Z2 Topological insulator
Protected by the mirror symmetry
of the crystal
Metallic state at its surface Protected
by time-reversal symmetry
Are there topological states protected by other symmetries?
Topological crystalline insulator (TCI)1
Metallic surface states
1. Fu, Liang., Physical Review Letters 106, 106802 (2011)
First material realization of topological crystalline insulator in SnTe
2. Hsieh, T. H., et al.,Nature Comm. , 3, 982 (2012)
3. Tanaka, Y., et al., Nature Physics 8, 800 (2012)
Theory2 Experiment3
Gresch et al, Phys. Rev. B 95, 075146 (2017)
Absence of a gap in the full xn(k) spectrum
nM =2 (For SnTe)
The individual Chern numbers C+i and C−i
defined on a mirror-invariant plane
Γ L1L2.
Strong indicator for the presence of a
topological phase
Chern number: N+ with H+ and N- with H-
Total Chern number: N++ N− = 0 (overall topology is trivial)
Mirror Chern number: nM= N+− N−
Teo, Fu, Kane Phys Rev. B 78, 045426 (2008)
(a) Brillouin zone of SnTe showing the mirror planes along which the xn’s are computed. (b) xn’s in the mirror plane. (c), (d) xn’s (circles) and their sum (rhombi) for the i and −i eigenstates.
Mirror Chern Number nM
xn
k
SnTe as a 3D crystalline topological insulator a
(001) surface Symmetries present: 1. Time reversal 2. Inversion 3. Reflection/Mirror
Surface states: SnTe (TCI): even number of Dirac points :Z2 (TI): odd number of Dirac points
Rocksalt structure
Mirror-symmetry plane ΓL1L2
Red: Sn Blue: Te
Band gap: Four equivalent L points in the FCC Brillouin zone Ordering of the conduction (derived from Te) and valence bands (from Sn) at L points is inverted relative to PbTe.
Sood et al: Experiments show a phonon anomaly in SnTe at ~ 1 GPa
• Electronic structure near the gap including spin-orbit coupling (SOI).
• There is crossing of bands among the conduction bands as we move from 0 Gpa to 2 GPa.
0 GPa 2 GPa
4 GPa
a) b)
c)
Effect of pressure on crystalline topological insulating state
0 5 10 15 20
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3 Valence Band Maxima
Conduction Bad Minima
En
erg
y (
eV)
Pressure (GPa)
EF
Increase in band gap at L point from 0 to 5 GPa
Evolution of band structure
Xn (circles) and their sum (rhombi) for the i and −i eigenstates in the mirror.
Ci =2 C-i=-2
Mirror Chern Number nM= 2
At 0 Gpa, 2GPa, 4 GPa.
The SnTe mirror Chern number calculated using the 'pw2z2pack' code, which calculates the overlap matrices for a specific symmetry eigenspace.
Z2 toplogical invariant=0 for kz=0 and kz=0.5 planes.
At 0 Gpa, 2GPa, 4 GPa.
Topological invariants 0 GPa
4 GPa 2 GPa
Band inversion between CBM and 2nd CBM of SnTe across the critical pressure is observed (at the L point).
0 GPa 2 GPa 3 GPa
Isosurfaces of charge density
Red: Sn Blue: Te
P-dependent transition seen in experiments (A K Sood et al): • Band inversion between CBM and 2nd CBM (at the L point) of SnTe
across the critical pressure (from 0 GPa to 2 Gpa).
• The mirror Chern number nM =2 and Z2 topological invariant =0 for SnTe at 0 GPa, 2 GPa, and 4 Gpa: No change in topology!
Summary (SnTe)
Raman: A Probe for Subtle Changes!
Sood et al, in preparation (2018)
Electronic Topological Phase Transitions
Topological Insulators
A cup can be deformed continuously into a donut: They have the same “Topological Invariant”
Nontrivial Topological Invariant of Electronic Structure → TI
Conduction Band
Conduction Band
TI Ordinary Insulator
Conduction Band TI
Surface of a TI
Vacuum
Surface Of a TI is Metallic
Prediction on As2Te3: Strain Induced Topological Insulator
Ordinary Insulator
Dirac Semimetal
Topological Insulator
Metallic Surface! Eg
Strain (%)
-6 -5 -4
TI
OI
Graphene-like 3D Electronic Structure
-5 % strain 0 % strain
Band Inversion
Prediction on As2Te3: Strain Induced Topological Insulator
Band Insulator
Topological Insulator
Metallic Surface!
Technological Applications
1. Stress Sensors (piezo-resistive) 2. Charge Pump 3. Thermoelectric
K Pal and U V Waghmare, App. Phys. Lett. 105, 062105 (2014)
Stress/Press
Current
Stress
Electronic Topological Transitions II Sb2Se3
e-phonon Coupling Broken Adiabaticity
Motivation • Identification of topological insulating phase relies on detecting the
gapless metallic surface states by ARPES. • Can we look for signatures of electronic topology in bulk property ? • Strategy: Look for a possibility of BI to TI transition under pressure and
signatures in bulk vibrational properties
3D Topological Insulators (TIs) Bi2 Se3
Bi2 Te3
Sb2 Te3
Band Insulator (BI) Sb2Se3 Space group : R-3m
Point group : D3d
??
Hexagonal unit cell
Zhang et al., Nature Phys. 5, 438 (2009)
Tune SOC (relative) by pressure
Rhombohedral structure of Sb2Se3
Li et al., arXiv: 1611.04688v1 (2016). Spin-orbit coupling (SOC) 44
Raman Shift Vs. Pressure Full width at half maximum
with Pressure
An anomalous increase of its line width by 200%
Experimental results
How do we understand these anomalies at 3 GPa ??
M1: Eg
M2: Ag M1: Eg
45
AK Sood et al
Band gap closes and reopens with pressure
Band gap with pressure
Electronic structure Band inversion at point
P < Pc
P > Pc
For P < Pc: 0 = 0 Band insulator For P > Pc: 0 = 1 Topological insulator
At Pc, the band gap vanishes (Dirac semimetal) and hence adiabatic approximation breaks down What happens to the vibrational properties at Pc?
Pc = 2 GPa
46
VBM
VBM
CBM
CBM
Phonon frequencies at different pressure
Adiabatic density functional theory
(Both frozen phonon approach and linear response theory yield frequencies within few cm-1 )
No anomaly in phonon frequencies observed at Pc = 2GPa
PUZZLE:
How to explain the observed phonon
anomalies at the ETT?
Evidence of strong electron-phonon coupling at Pc
from first-principles
Change in electronic structure with structural change Associated with A1g mode:
Distorted (solid line)
Undistorted (dotted line)
Atomic displacements of a phonon mode are frozen and structure is distorted
E (
eV
)
For A2u mode:
Undistorted (dotted line)
Distorted (Solid line)
Each of the doubly degenerate valence and conduction bands split and minima or maxima of the bands shift away from Gamma point.
E (
eV
)
Breakdown of the adiabatic approximation at P ~ Pc
Dynamical corrections to phonon frequencies
Phonon self energy:
𝑔𝑘𝑖,𝑘𝑗 = < 𝑘𝑖 𝐻𝐴1𝑔 𝑘𝑗 > for 𝐴1𝑔 mode
= < 𝑘𝑖 𝐻𝐴2𝑢 𝑘𝑗 > for 𝐴2𝑢 mode
= < 𝑘𝑖 𝐻𝐸𝑔 𝑘𝑗 > for 𝐸𝑔 mode
𝐻 𝐸𝑔= 𝐴𝐸𝑔[(𝑢𝐸𝑔𝑥 𝛤15 + 𝑢𝐸𝑔
𝑦𝛤25)𝛤35 - h.c ] /2i
𝐻𝐴2𝑢= 𝐴𝐴2𝑢𝑢𝐴2𝑢𝛤45 𝐻𝐴1𝑔= 𝐴𝐴1𝑔𝑢𝐴1𝑔𝛤5
EPC Hamiltonian for different phonon mode (𝜈)
𝛤𝑚 = Dirac matrix 𝛤𝑚𝑛 = [𝛤𝑚,𝛤𝑛]/2i
𝑔𝑘𝑖, 𝑘+𝑞 𝑗𝜈 = Matrix element of electron-phonon coupling(EPC) for 𝜈 phonon mode
51
Pc P<Pc P>Pc P<Pc P>Pc Pc
Raman active modes show anomaly at the transition pressure Pc = 2GPa
Line-width due to electron-phonon coupling
FWHM
Eg M. Lazzeri etl al., Phys. Rev. B, 73 155426 (2006)
M0 = -k(P-Pc)
52 Bera et al, Phys. Rev. Lett. 110, 107401 (2013)
Conclusions
• First-principles calculations confirm the observed ETT through electronic band inversion across the gap.
• Electron-phonon coupling near Pc requires us to go beyond the
adiabatic Born-Oppenheimer approximation and include dynamical correction to phonon frequencies.
Analysis with 4-band k.p model gives a qualitatively correct description of the experimental anomalies
53 Bera et al, Phys. Rev. Lett. 110, 107401 (2013)
(under revision, NPJ Quantum Materials) 54
Dirac semi-metallic State (DSM)
Transition (critical) state of BI –TI transition
Consequence of Cn rotational (n=3, 4, 6)
Cava et al., Phys. Rev. B 91, 205128 (2015) Nagaosa et al., Nature Comm. 5, 4898 (2014);
(Sb2Se3, -As2Te3 etc.)
(Na3Bi : C3 symmetry, Cd3As2 : C4 symmetry)
Motivation
Can we achieve a DSM state in non-centrosymmetric materials?
DSM state is observed mainly in centro-symmetric material e.g., Na3Bi, Cd3As2
55
Sheet et al, Nature materials 15, 32 (2016)
LiMgBi
Li : (0, 0, 0)
Mg : ( ½, ½, ½ )
Bi : (¼, ¼, ¼ ) Bi Mg
ZB: Zinc-blende sub-lattice RS: Rock-salt sub-lattice
Tunability of topological semi-metallic states We can apply C3-rotataional symmetry preserving strain in ternary HHs Intuition: We may get a DSM out of HHs
A large number of half-Heusler (non-cetrosymmetric) compounds show topological semi-metallic states
MZ Hasan et al, Nature Materials 9, 546 (2010)
Li
56
Electronic structure of LiMgBi
Zhang et al., Nature Materials, 9, 541 (2010).
Proto-typical example of a semimetal with non-trivial electronic topology
Topological semi-Metal in the Native State
Our work: Strain preserving C3 symmetry along <111>
We focus on LiMgBi: Light-weight Half-Heusler and show that results are general to HH’s! 57
Tensile strain along [111] direction of LiMgBi: Dirac cone along three-fold axis
Dirac cone along -L direction The C3 rotational axis, Dirac ones at k0<111> and -k0<111>
58
There is another Dirac cone present along -(-L) direction.
Topological Phase Diagram of LiMgBi
In a single compound, applying different combination of strains, we get different phases of matter!
TDSM
TI
BI
TSM
DSM 59
Generic Topological Phase Diagram(s) of Half-Heuslers
60
Dirac points along [111] and –[111]
directions in the TDSM state
Γ
[111]
Anisotropy (Bands split once they deviate from [111] direction) Chern numbers of the inner and outer Fermi spheres are opposite Nontrivial Berry phases for orbits
E
k
Pal & Waghmare (2018)
Summary: DSMs in Half Heuslers
We have presented generic topological phase diagram of half-Heusler (HH) compounds. We show that strained structures of HHs host topological insulating, topological semi-metallic, Dirac semi-metallic (DSM) and band insulating states. The DSM state in LiMgBi is stabilized by C3 rotational symmetry. .
62
Thermoelectrics
We considered the following class of materials
• Many good thermoelectric (TE) materials came to be known as topological insulators. Examples: Bi2Se3, Bi2Te3, SnTe etc.
• We investigated the connection between topology and TE efficiency
Motivation
TI = Topological insulators TCI = Topological crystalline insulators TSM = Topological semimetal DSM = Dirac semimetal WSM = Weyl semimetal
Bejenari et al., Phys. Rev. B, 78, 115322 (2008)
Thermoelectric power factor (P): S2 64
Table summarizing TE performance of all the materials studied
-As2Te3 and TaAs exhibit good TE power factor which is comparable to that of PbTe
65
Electronic structures
As band inversion introduces multiple band extrema, TI phases should exhibit better TE performance
Multiple band extrema Valence band convergence Asymmetry in DOS near EF
Features in the electronic structure that boost TE performance
66
What do we learn?
Pal, Anand, Waghmare, J Mat Chem 3, 12130 (2015)
WTI Z2-TI
BiSe (a Weak TI) as a thermoelectric
Experiments: K Biswas et al
BiSe: Smaller gap, softer phonons
BiSe
BiSe Bi2Se3
Samanta et al, J. Am. Chem. Soc. 2018, 140, 5866−5872
Ultra-low thermal conductivity: dissipation of heat into soft optic modes of Bi2
Summary: thermoelectric properties of TI’s and DSM’s
• Topological insulators with small band gaps are excellent thermoelectrics, while Dirac semimetals may not be quite so. • High TE performance: - Band inversion in an electronic topological transition giving many local extrema in the valence and conduction bands - Heavy atoms, weak bonds: soft phonons giving low kthermal
• We predict that Weyl semimetal TaAs is a good TE material and should be explored experimentally.
69
Summary
Geometric Phases: Anomalous Transport & Electronic Topology Dirac Semi-Metal as a Parent phase of TIs, Weyl semi-metals, … Prediction of Half-Heuslers e-phonon coupling: Broken Adiabaticity at ETT > Raman as a powerful tool Potential Applications of Thermoelectric Properties
Acknowledgements
Koushik Pal, S Anand (Northwestern) Raagya Arora Sharmila Shirodkar (Rice) J Bhattacharjee (NISER)
A K Sood* and His Group (IISc )
K Biswas and His Group (JNCASR )
Funding from: JC Bose Fellowship of DST and IKST
Thank you!