Umesh V Waghmare - TIFRQmat/umesh waghmare_qmat-2018.pdf · Umesh V Waghmare Funded by Dept of Sci and Tech, Govt of India . Talk is focused on

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!