Topological insulators and superconductors

KITPC 2010Shoucheng Zhang, Stanford University

Colloborators

Stanford group: Xiaoliang Qi, Andrei Bernevig, Congjun Wu, Chaoxing Liu, Taylor Hughes, Sri Raghu, Suk-bum Chung

Stanford experimentalists: Yulin Chen, Ian Fisher, ZX Shen, Yi Cui, Aharon Kapitulnik, …

Wuerzburg colleagues: Laurens Molenkamp, Hartmut Buhmann, Markus Koenig, Ewelina Hankiewicz, Bjoern Trauzettle

IOP colleagues: Zhong Fang, Xi Dai, Haijun Zhang, …

Tsinghua colleagues: Qikun Xue, Jinfeng Jia, Xi Chen,…

Outline

Models and materials of topological insulators

General theory of topological insulators, exotic particles

Topological superconductors

The search for new states of matter

Crystal: Broken translational symmetry

Magnet: Broken rotational symmetry

Superconductor: Broken gauge symmetry

The search for new elements led to a golden age of chemistry.

The search for new particles led to the golden age of particle physics.

In condensed matter physics, we ask what are the fundamental states of matter?In the classical world we have solid, liquid and gas. The same H2O molecules can condense into ice, water or vapor.

In the quantum world we have metals, insulators, superconductors, magnets etc.Most of these states are differentiated by the broken symmetry.

The quantum Hall state, a topologically non-trivial state of matter

h

enxy

2

• TKNN integer=the first Chern number.

)()2( 2

2

kFkd

n

• Topological states of matter are defined and described by topological field theory:

AAxdtdS xy

eff 2

2

• Physically measurable topological properties are all contained in the topological field theory, e.g. QHE, fractional charge, fractional statistics etc…

• von Klitzing, 1980

HgTe Theory: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experiment: Koenig et al, Science 318, 766 (2007)BiSb Theory: Fu and Kane, PRB 76, 045302 (2007)Experiment: Hsieh et al, Nature 452, 907 (2008)Bi2Te3, Sb2Te3, Bi2Se3 Theory: Zhang et al, Nature Physics 5, 438 (2009)Experiment Bi2Se3: Xia et al, Nature Physics 5, 398 (2009), Experiment BieTe3: Chen et al Science 325, 178 (2009)On average 2-3 paper per day on the subject!

Discovery of the 2D and 3D topological insulator

Topological Insulator is a New State of Quantum Matter

From traffic jam to info-superhighway

on chip

Traffic jam inside chips today Info highways for the chips in the future

Quantum Hall effect and quantum spin Hall effect

Topological protection (Qi and Zhang, Phys Today, Jan, 2010)

Spin=1/2

-

The topological distinction between a conventional insulator and a QSH insulator

Kane and Mele, Wu, Bernevig and Zhang; Xu and Moore

• Band diagram of a conventional insulator, a conventional insulator with accidental surface states (with animation), a QSH insulator (with animation). Blue and red color code for up and down spins.

k

Trivial TrivialNon-trivial

k=0 or

5 .1 5 .2 5 .3 5 .4 5 .5 5 .6 5 .7 5 .8 5 .9 6 .0 6 .1 6 .2 6 .3 6 .4 6 .5 6 .76 .6

1 .0

1 .5

0 .5

0 .0

-0 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

5 .0

6 .0

5 .5

B a n d g a p v s . la ttic e co n sta n t(a t ro o m te m p e ra tu re in z in c b le n d e s t ru c tu re )

Ban

dga

p e

nerg

y (e

V)

la ttice con sta n t a [Å ]0

Band Structure of HgTe

S

P

S

P3/2

P1/2

S

P SP1/2

P3/2

CdTe

HgTe

CdTe

E1

H1

normal

HgTe

CdTe CdTe

H1

E1

inverted

Band inversion in HgTe leads to a topological quantum phase transition

Let us focus on E1, H1 bands close to crossing point

The model of the 2D topological insulator (BHZ, Science 2006)

),(,,(,,,, yxyx ippippss

Square lattice with 4-orbitals per site:

Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k.

)(0

0)(),(

kh

khkkH yxeff

2

2

)(

)(

)()()sin(sin

)sin(sin)()(

BkmikkA

ikkABkm

kdkmkikA

kikAkmkh

yx

yx

aa

yx

yx

Similar to relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! m/B<0 for d>dc.

kx

E

Mass domain wall

Cutting the Hall bar along the y-direction we see a domain-wall structure in the band structure mass term. This leads to states localized on the domain wall which still disperse along the x-direction, similar to Jackiw-Rebbi soliton.

m/B<0 m

y

0

y

x

x

Bulk

Bulk

E

0

Experimental setup

• High mobility samples of HgTe/CdTe quantum wells have been fabricated.• Because of the small band gap, about several meV, one can gate dope this system from n to p doped regimes.• Two tuning parameters, the thickness d of the quantum well, and the gate voltage.•(Koenig et al, Science 2007)

Experimental observation of the QSH edge state (Konig et al, Science 2007)

x

x

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

R (

k)

V* (V)

I: 1-4

V: 2-3

1

3

2

4

R14,23=1/4 h/e2

R14,14=3/4 h/e2

Nonlocal transport in the QSH regime, (Roth et al Science 2009)

No QSH in graphene

• Bond current model on honeycomb lattice (Haldane, PRL 1988).

• Spin Hall insulator (Murakami, Nagaosa and Zhang 2004)

• Spin-orbit coupling in graphene (Kane and Mele 2005). They took atomic spin-orbit coupling of 5meV to estimate the size of the gap.

• Yao et al, Min et al 2006 showed that the actual spin-orbit gap is given by 2

so/=10-3 meV.

• Similar to the seasaw mechanism of the neutrino mass in the Standard Model!

Single valley 2D Dirac cone

• At the critical thickness, the BHZ model reduces to the massless Dirac model in 2+1d, with a single valley.

• Recent experiments on HgTe has reached this critical point!

• HgTe provides an ideal platform to study 2D massless Dirac fermions!

(b)

Se2

BiSe1

Quintuplelayer

B

B

CA

C

A

C

(a)

t1

t2 t3

z

x

y

(c)x

y

3D insulators with a single Dirac cone on the surface

(I) (II) (III)

(a)

Bi

Se

(eV)

E (

eV)

-0.2

0.2

0.6

0 0.2 0.4

(b)

c

Relevant orbitals of Bi2Se3 and the band inversion

(a) Sb2Se3 (b) Sb2Te3

(c) Bi2Se3 (d) Bi2Te3

Model for topological insulator Bi2Te3, (Zhang et al, 2009)

Pz+, up, Pz-, up, Pz+, down, Pz-, down

Single Dirac cone on the surface of Bi2Te3

Surface of Bi2Te3 = ¼ Graphene !

Doping evolution of the FS and band structureDoping evolution of the FS and band structure

EF(undoped)BCB bottom

Dirac point position

Undoped Under-doped Optimally-doped Over-doped

BVB bottom

Arpes experiment on Bi2Te3 surface states, Shen group

Arpes experiment on Bi2Se3 surface states, Hasan group

Surface states of TIs vs Rashba SOC

3D TI

Surface Rashba term: Breaking of inversion symmetry at the surface.

Unlike graphene, here two components are related by time reversal, and Pauli matrix is the real spin.

Strong SOC

Conduction band

Valence band

E EConduction band

Valence band

Spin-plasmon collective mode (Raghu, Chung, Qi+SCZ, PRL2009)

• General operator identity:

• Density-spin coupling:

Surface Landau levels, Xue group

General theory of topological insulators

• Topological band theory based on Z2 topological band invariant of single particle states.(Fu, Kane and Mele, Moore and Balents, Roy)

• Topological field theory of topological insulators. Generally valid for interacting and disordered systems. Directly measurable physically. Relates to axion physics! (Qi, Hughes and Zhang)

• For a periodic system, the system is time reversal symmetric only when=0 => trivial insulator= => non-trivial insulator

Generalization of the QH topology state in d=2 to time reversal invariant topological state in d>2, in Science 2001

The periodic table of topological states:(Qi, Hughes and Zhang, Ludwig et al, Kitaev)

The TRI topological insulators form a dimensional chain: 4D=>3D=>2D

The 4D TRI topological insulator directly inspired the discovery of the intrinsic spin Hall effect in d=3, Science 2003

“Recently, the QHE has been generalized to four spatial dimensions (4). In that case, an electric field induces an SU(2) spin current through the nondissipative

transport equation…The quantum Hall response in that system is physically realized though the spin-orbit coupling in a time reversal symmetric system.”

TRB topological insulators in d=2• Chern-Simons topological field theory, odd under TR:

• 1st Chern number

TRI topological insulators in d=4

• Chern-Simons topological field theory, even under TR:

• 2st Chern number

• If we replace k4 by an adiabatic parameter, we obtain the quantized cyclic change of the magneto-electric polarization in d=3!

• If we perform Kaluza-Klein compactification, we obtain the effective field theory of the d=3 topological insulator!

The best way to understand TRI TI is D=4 => D=3 => D=2

The best way to understand critical phenomenon is D=4 => D=4-

Dimensional reduction• From 4D QHE to the 3D topological insulator

• From 3D axion action to the 2D QSH

FFtxxdtdS

FFtxAdxxdtd

FFAxdtdS

D

DQH

),(

)),((

33

553

44

AxdtdS

AtxdzAxdtd

AAxdtdS

D

z

D

22

2

33

)),((

2

2

1

22

eJ

eJ

D

D

Goldstone & Wilzcek

Zhang & Hu, Qi, Hughes & Zhang

(x, y, z)

x5

A5

TRI topological insulators in d=3

• Axion field theory, even under TR only when =0, :

• Magneto-electric polarization (QHZ 2008):

Generalization to general interacting TI (Wang, Qi, SCZ)

• Topological order parameter for generally interacting TI• Experimentally measurable through the topological magneto-electric effect• WZW extension u introduces integer ambiguity of P3• P3 is topologically quantized to be integer or half-integer• Also applies to disordered systems, see Li et al, Groth et al.

Topological field theory and the family tree• Topological field theory of the QHE: (Thouless et al, Zhang, Hansson and Kivelson)

• Topological field theory of the TI: (Qi, Hughes and Zhang, 2008)

)()(..))()(( 43 xdAxdAxdkdakakdS

)()()( 32 xdAxAxdkdakdS

• More extensive and general classification soon followed (Kitaev, Ludwig et al)

TRI topological insulator d=2, characterized by the discrete Z2 topological number in 2005

Topological band theory

The continuum TI formula and the discrete TI formula are pre-destined to converge! (Wang, Qi and Zhang, 0910.5954)

Equivalence between the integral and the discrete topological invariants (Wang, Qi and Zhang, 0910.5954)

LHS=QHZ definition of TI, RHS=FKM definition of TI

RKKY coupling of the surface states (Liu et al, PRL 2009)

Low frequency Faraday/Kerr rotation(Qi, Hughes and Zhang, PRB78, 195424, 2008, Zhang group 2010, MacDonald group

2010)

Adiabatic

Requirement:

(surface gap)

Universal quantization in units of the fine structure constant!

Seeing the magnetic monopole thru the mirror of a TME insulator, (Qi et al, Science 323, 1184, 2009)

TME insulator

q

(for =’, =’)similar to Witten’s dyon effect

higher order feed back Magnitude of B:

Dynamic axions in topological magnetic insulators(Li et al, Nature Physics 2010)

• Hubbard interactions leads to anti-ferromagnetic order

• Effective action for dynamical axion

Axions and dark matter on your desktop? (Zhang group, Nature Physics 2010)

Now Shou-Cheng Zhang and his colleagues inform us that, all

along, axions have been lurking unrecognized on surfacesof topological insulators.

Frank Wilzcek, NATURE 458, 129 (2009)

Topological insulators and superconductors

Full pairing gap in the bulk, gapless Majorana edge and surface states

Chiral Majorana fermions

Chiral fermions

massless Majorana fermions

massless Dirac fermions

Qi, Hughes, Raghu and Zhang, PRL, 2009

Topological superconductors and superfluids

The BCS-BdG model for 2D equal spin pairing model of 2D TI by BHZ

where p+=px+ipy. The edge Hamiltonian is given by:

forming a pair of Majorana fermions. Mass term breaks T symmetry=> topological protection!

Qi, Hughes, Raghu and Zhang, PRL, 2009 Schnyder et al, PRB, 2008 KitaevRoy

Tanaka, Nagaosa et al, PRB, 2009 Sato, PRB, 2009

Probing He3B as a topological superfluid (Chung

and Zhang, 2009) The BCS-BdG model for He3B Model of the 3D TI by Zhang et al

Surface Majorana state:

Qi, Hughes, Raghu and Zhang, PRL, 2009 Schnyder et al, PRB, 2008 KitaevRoy

Chiral superconductor obtained from QAH+SC (Qi,

Hughes and SCZ, 2010)

General reading

For a video introduction of

topological insulators and superconductors,

see http://www.youtube

.com/watch?v=Qg8Yu-Ju3Vw

Summary: discovery of new states of matter

Magnet s-wave superconductor

Crystal

Quantum Spin Hall Topological insulators