Realization of three-dimensional and two-dimensional topological

113_6Liu_Lyon2009.pdf

Realization of three-dimensional and

two-dimensional topological insulators in B 2 S 3 type of materials (B=Bi, Sb and S=Se, Te)

Chao-xing Liu

Physical Institute (EP3) and Institute for Theoretical Physics and

Astrophysics,University of Wuerzburg, Wuerzburg

Collaborators: X.L. Qi, S.C. Zhang (Stanford), H.J. Zhang, X.Dai, Z. Fang (IOP), B.H. Yan, T. Frauenheim (Bremen) Nature Physics 5, 438 (2009). Cond-mat/0908.3654 (2009)

Introduction

Basic property of topological insulator

Introduction

Basic property of topological insulator

H Topological insulator has bulk gap and gapless

surface or edge states which are protected by the bulk topology.

Introduction

Basic property of topological insulator

H Topological insulator has bulk gap and gapless

surface or edge states which are protected by the bulk topology. H TRI TIs are induced by strong spin-orbit coupling and preserve time reversal symmetry.

Introduction

Basic property of topological insulator

H Topological insulator has bulk gap and gapless

surface or edge states which are protected by the bulk topology. H TRI TIs are induced by strong spin-orbit coupling and preserve time reversal symmetry.

H TRI TI can be realized in both 2D and 3D.

Introduction

Basic property of topological insulator

H Topological insulator has bulk gap and gapless

surface or edge states which are protected by the bulk topology. H TRI TIs are induced by strong spin-orbit coupling and preserve time reversal symmetry. H TRI TI can be realized in both 2D and 3D. H Experimentally, both 2D and 3D TIs have been discovered.

2D: HgTe/CdTe QWs

Bernevig, et al, Science 314, 1757 (2006)Koenig, et al, Science 318, 766 (2007)

3D: Bi

x Sb 1-x L. Fu, et al, PRB 76, 045302 (2007)Hsieh, et al, Nature 452, 970 (2007)

Introduction

New 3D topological insulator material

H A new class of 3D topological insulators: B 2 S 3 (B=Bi, Sb and S=Se, Te)

H.J. Zhang et al, Nat Phys 5, 438 (2009) Y. Xia et al, Nat Phys 5, 398 (2009).Y. L. Chen et al, Science 325, 178 (2009).

Introduction

New 3D topological insulator material

H A new class of 3D topological insulators: B 2 S 3 (B=Bi, Sb and S=Se, Te)

H.J. Zhang et al, Nat Phys 5, 438 (2009) Y. Xia et al, Nat Phys 5, 398 (2009).Y. L. Chen et al, Science 325, 178 (2009).

H What we will discuss in this talk

•How to realize 3D topological insulator in realistic materials: Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 •The relation between 3D and 2D topological insulators. Possible realization of 2D topological insulator with Bi 2 Se 3 , Bi 2 Te 3 thin film.

3D topological insulator: B

2 S 3 (B=Bi, Sb and S=Se, Te)

Nature Physics 5, 438 (2009)

Band dispersion of Bi

2 Se 3 3D TI Without spin-orbit couplingWith spin-orbit coupling

Band dispersion of Bi

2 Se 3 3D TI Without spin-orbit couplingWith spin-orbit couplingAnti-crossing

Band dispersion of Bi

2 Se 3 3D TI Without spin-orbit couplingWith spin-orbit couplingAnti-crossing

HWhat is the origin for this anti-crossing?

HWhat is the physical consequence of this anti-

crossing?

Inversion

center

Crystal structure of Bi

2 Se 3 3D TI

HLayered structure with five atomic

layers as one quintuple layer

HSe2 is an inversion center.

States at Γ point has definite parity

Atomic orbitals of Bi

2 Se 3 3D TI

HThe states near gap are dominant by p orbital.

Without spin-orbit coupling

HThe parities for conduction band and

valence band are opposite. P1 + α is a bonding state of Bi and Se p orbitals while P2 - α is an anti-bonding state. BiSe

Spin-orbit coupling

3D TI

Spin-orbit coupling

3D TI

Spin-orbit coupling

3D TILevel repulsionLevel repulsion

λ 0 atomic SOC

Band inversion induced by SOC

3D TIWithout SOCWith SOC

•Anti-crossing comes from the strong SOC of Bi. •What's the physical consequence of this anti- crossing?

Topological insulator phase

3D TI H Z 2 topological classification and Parity rule

Fu and Kane (2007)

HEffective Model: H Topological insulator phase with Dirac type surface states. • To get a better understanding, in the following we will construct the effective model in the basis

Effective model

3D TI 2M 0 >0 Gap

Effective model

3D TI H

Inverted regime: M

0 <0. 2M 0 <0 Gap

Effective model

3D TI H

Inverted regime: M

0 <0. 2M 0 <0 H

Semi-metal: coupling between P1

+ z and P2 - z is needed to open a gap. Gap

Effective model

3D TI

HWhat kind of coupling is possible?

Effective model

3D TI

HWhat kind of coupling is possible?

• Opposite parity indicates k linear coupling is the lowest order.

Effective model

3D TI

HWhat kind of coupling is possible?

• Opposite parity indicates k linear coupling is the lowest order.• Total angular momentum

Effective model

3D TI

HWhat kind of coupling is possible?

• Opposite parity indicates k linear coupling is the lowest order.• Total angular momentum term forand

Effective model

3D TI

HWhat kind of coupling is possible?

• Opposite parity indicates k linear coupling is the lowest order.• Total angular momentum term forandterm forand

Effective model

3D TI

HWhat kind of coupling is possible?

• Opposite parity indicates k linear coupling is the lowest order.• Total angular momentum term forandterm forand

Effective model

3D TIEnergy dispersion:

Effective model

3D TI

HQuite similar to relativistic 3D Dirac equation

Energy dispersion:

Effective model

3D TI HQuite similar to relativistic 3D Dirac equationHDifference:• Quadratic term • Negative mass, M 0 <0

Energy dispersion:

Theory of invariant

3D TI HHamiltonian need to be invariant under the symmetry operation of crystal.

Any four band model:

Γ matrix:Basis: Eg. Rotation along z axis: behavior as a vector as (k x , k y ). is one of possible term.

Theory of invariant

3D TI

HConstruct the Hamiltonian with

character table of group

Theory of invariant

3D TI

HConstruct the Hamiltonian with

character table of group

Lattice symmetry

3D TI:

Lattice symmetry

3D TIC

3 :invariant under C 3 rotationABand odd parity:

Lattice symmetry

3D TIC

3 :invariant under C 3 rotationABand odd parity

Hk cubic term breaks full rotation symmetry

down to C 3 symmetry. :

Surface states

3D TI Bi 2 Se 3 z vacuum z=0 H A surface state with a single Dirac cone at the Γ point will appear for Bi 2 Se 3 when there is a boundary.

Surface states

3D TI Bi 2 Se 3 z vacuum z=0 H A surface state with a single Dirac cone at the Γ point will appear for Bi 2 Se 3 when there is a boundary. HAt finite Fermi energy, spin polarization forms a vortex configuration (Rashba type).

Surface states as domain wall fermion

3D TI H

Mass M

0 domain wall zM z<0 M 0 >0 z>0 M 0 <0 TI z M 0 <0M 0 >0vacuum z=0Vacuum is viewed as

HKrammer pair of two localized states

The localized states are determined by the sign change of the mass and not sensitive to how the sign is changed.

JPSJ, 77, 031007 (2008)

Effective Hamiltonian for Surface states

3D TI HHamiltonian can be projected into the sub-space of two localized states

Single Dirac cone Hamiltonian:

L. Fu, cond-mat/0908.1418

HRashba type term

Breaking of inversion symmetry due to

the surface, however the in-plane rotation symmetry is preserved. Hk 3 term z direction spin polarization with six fold symmetry, breaking the in-plane rotation symmetry. 3D TI HThis surface state is a direct generalization of helical edge states of 2D TI with quantum spin Hall effect.

3D topological insulator

yzxxyk-k HThe effective Hamiltonian and the surface states have shown that Bi 2 Se 3 is a 3D topological insulator.

Guiding principle for searching new TI

materials 3D TI

HStrong spin-orbit coupling system.

Eg. HgTe, Bi

x Sb 1-x , Bi 2 Se 3

HSmall gap systemHAtomic orbital

Eg. s-p orbital coupling: HgTe; p-p orbital coupling: Bi x Sb 1-x , Bi 2 Se 3 ; d orbital coupling: Na 2 IrO 3 , A 2 Ir 2 O 7 ; • The gap should be possible to be inverted by SOC, therefore can not be too large. • Electronegativity for the atoms in the compound should be similar and the type of bonding should be more covalent rather than ionic.

2D Quantum spin Hall phase in Bi

2 Se 3 and Bi 2 Te 3 thin film

C.X. Liu, et al, Cond-mat/0908.3654, (2009)

H2D QSH insulator

Materials for 2D QSH insulator and 3D

topological insulator

2D QSHI

HgTe/CdTe quantum wells: ~ 40meV

H3D topological insulator

H2D QSH insulator

Materials for 2D QSH insulator and 3D

topological insulator

2D QSHI

HgTe/CdTe quantum wells: ~ 40meV

H3D topological insulator

Bi x Sb 1-x , Bi 2 Se 3 , Bi 2 Te 3 , Sb 2 Te 3 Bi 2 Se 3 : ~300meV

HBand inversion

Similarity between 2D QSHI material

and 3D TI material

2D QSHIHgTe/CdTe

quantum well Bi 2 Se 3

Topological phase transition occurs when

there is a band inversion.

HEffective Hamiltonian

Similarity between 2D QSHI and 3D TI

2D QSHI

3D TI

Hamiltonian

2D QSH

Hamiltonian

HCan we realize 2D QSH insulator with 3D TI?

3D TI material

(Bi 2 Se 3 or Bi 2 Te 3 )

Normal insulator

or vacuum d

3D TI with confinement

2D QSHI

Normal insulator

or vacuum

H3D TI material in quantum well configuration

H

In the following, we first consider the A

1 =0 case and then turn on A 1 .

Energy level in quantum well

2D QSHI

case: H k z term only appear in diagonal term M(k), so it is simply an infinite quantum well problem. H A series of crossing points appear due to the inverted bulk band. H Phase transition occurs at H n and E n states due to opposite parities. Similar to HgTe/CdTe Quantum well system

Energy level in quantum well

2D QSHI

case: H k z term only appear in diagonal term M(k), so it is simply an infinite quantum well problem. H A series of crossing points appear due to the inverted bulk band. H Phase transition occurs at H n and E n states due to opposite parities. Similar to HgTe/CdTe Quantum well system

Energy level in quantum well

2D QSHI

QSH phase

case: H k z term only appear in diagonal term M(k), so it is simply an infinite quantum well problem. H A series of crossing points appear due to the inverted bulk band. H Phase transition occurs at H n and E n states due to opposite parities. Similar to HgTe/CdTe Quantum well system

Energy level in quantum well

2D QSHI

case: H Due to parity, there is still crossing between E n and H n states. H There is coupling between E n and H n+1(n-1) , which induce the anti- crossing. Quantum spin Hall phase still exist!

QSH phase

Energy level in quantum well

2D QSHI

case: H Due to parity, there is still crossing between E n and H n states. H There is coupling between E n and H n+1(n-1) , which induce the anti- crossing. Quantum spin Hall phase still exist!

QSH phase

Energy level in quantum well

2D QSHI

case: H Due to parity, there is still crossing between E n and H n states. H There is coupling between E n and H n+1(n-1) , which induce the anti- crossing. Quantum spin Hall phase still exist!

QSH phase

Energy level in quantum well

2D QSHI

?case: H Due to parity, there is still crossing between E n and H n states. H There is coupling between E n and H n+1(n-1) , which induce the anti- crossing. Quantum spin Hall phase still exist!

QSH phase

H New states S1 and S2 always stay in the middle of the gap.

Surface states picture

2D QSHI

H States S 1 and S 2 are the bonding and anti-bounding states of two surface states at opposite surfaces. H The change of the sequence of S 1 and S 2 will induce a topological phase transition between QSH phase and normal insulator.

Surface states picture

2D QSHI

H States S 1 and S 2 are the bonding and anti-bounding states of two surface states at opposite surfaces.

Hybridization

H The change of the sequence of S 1 and S 2 will induce a topological phase transition between QSH phase and normal insulator.

Surface states picture

2D QSHI

H States S 1 and S 2 are the bonding and anti-bounding states of two surface states at opposite surfaces.

Hybridization

H The change of the sequence of S 1 and S 2 will induce a topological phase transition between QSH phase and normal insulator.

Hybridized gap

Realistic materials Bi

2 Se 3 and Bi 2 Te 3 thin film

2D QSHI

H Bi 2 Se 3 type of materials have layered structure, suitable for thin film growth. HAb initio calculation indeed show the oscillation between normal insulator and QSH insulator. Bi 2 Se 3 Bi 2 Te 3

Recent experiments

2D QSHI

HHybridized gap is observed recently.

HFurther transport measurement is needed to prove the oscillation between different phases.

Y. Zhang et al, arXiv:0911.3706 (2009)

•Here we show how to realize the 3D and

2D topological insulators in Bi

2 Se 3 type of materials. •The experience we learn here can be used to search for new 2D and 3D topological insulators.

Summary