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Topological)Insulators)a"beginner’s"guide)

Luis)Morellón)Ins4tute)of)Nanoscience)of)Aragón)(INA))

Zaragoza,)Spain)

))

!

!

Luis)Morellon) Topological)Insulators)

MARTES)CUÁNTICO)FAC.)CIENCIAS,)UNIZAR)24/02/2015)

))Outline) ))

Luis)Morellon)

Topological!Insulators!(TI’s)!(a)beginner’s)guide))

o !The!Quantum!Hall!Effect!(QHE)!!!!!(integer))

o !2D!TI’s!

o !3D!TI’s!

o !Progress!on!TI’s!

Topological)Insulators)

Insulating State

E

k

EG

(a) (b) (c)

0 /a−π/a−π

Most)basic)state)of)maPer:)insula4ng)state.)Simplest)case:)atomic)insulator)(solid)Ar))

))Outline) ))

Luis)Morellon)



o !2D!TI’s!

o !3D!TI’s!



VOI vM+ $5s +vMQ&R PHYSIC:AI. REVIEW LETTERS 11 AvGvsY 1980

ew et od for High-Accuracy Determination f th F -So e ine- tructure ConstantBased on Quantized Hall Resistance

K. v. KlitzingHsysikalisches Institut der Universitat Wurzburg, D-8700 ~iirgburg, Federal Re b

IIochfeld-Ma gn etlabor des Max-Planck -Ins tituturgburg, I'ederal Republic of Germany, and

x- anc - nstituts pier PestkorPerforsckung, P 38048-Grenoble, Prance

G. DordaForschungslaboratorien der Siemens AG, D-8000 Mun0 uncken, Pedera/ RePublic of Germany

and

M. PepperCavendish Laboratory, Cambridge CB30HZ Unoted Kingdom

(Received 30 May 1980)

Measurements of the Hall voltage of a two-di'

1 I~

]0 ~ ~

wo- imensiona electron gas, realized with asi icon metal-oxide-semiconductor field-effect transistor, show that the Hall resiat particular, experimentall well-d

ow a e a resistance

which de end only we - e ined surface carrier concentrations h f' d

p y on the fine-structure constant and speed of li ht d'as ixe va ues

the come trg ry of the device. Preliminary data are reported.~ ~ ~

o ig, an is insensitive to

PACS numbers: 73.25.+i, 06.20.Jr, 72.20.My, 73.40.Qv

In this paper we report a new, potentially high-

accuracy method for determining the fine-struc-ture constant, n. The new approach is based on

the fact that the degenerate electron gas in the in-

version layer of a MOSFET (metal-oxide-semi-conductor field-effect transistor) is fully cluan-

tized when the transistor is operated at heliumtemperatures and in a strong magnetic field of

order 15 T.' The inset in Fig. 1 shows a schem-atic diagram of a typical MOSFET device used in

this work. The electric field perpendicular to thesurface (gate field) produces subbands for the mo-tion normal to the semiconductor-oxide interface,and the magnetic field produces Landau quantiza-

tion of motion parallel to the interface. The den-

sity of states D(E) consists of broadened 5 func-

tions'; minimal overlap is achieved if the mag-netic field is sufficiently high. The number of

states, NL, within each Landau level is given by

V„=ea/I, (&)

UHI N

li

25 -2.5

20.-2.0

15-1.5

10 -1.0

5--0.5

0;0:;

Upp lmVp-SUBSTRATE

HALL PROBE

10 15 20

--ORAIN

~ ~ g SURFACE CHANNEL $~&n'

SOURCE GATE

/POTENTIAL PROBES

25

where we exclude the spin and valley degenera-cies. If the density of states at the Fermi ener-

gy, N(EF), is zero, an inversion layer carriercannot be scattered. , and the center of the cyclo-tron orbit drifts in the direction perpendicular tothe electric and magnetic field. If N(FF) is finitebut small, an arbitrarily small rate of scatteringcannot occur and localization produced b th llxf t

y e ong

e arne is the same as a zero scattering rate,i.e. , the same absence of current-carrying statesoccurs. ' Thus, when the Fermi level is between

n=Q -n=l n=2

= Vg/V

FIG. l. Recordings of the Hall voltage U and th

volH, an e

ftage drop between the potential prob Uo es, &&, asa

unction of the gate voltage V at T =1.5 K. The con-stant magnetic field B) is 18 T and the source draincurrent, l, is 1 A.p, . The inset shows a top view of thedevice with a length of I =400 pm, a width of 8' =50 pm,and a distance between the potential probes f Ip,m.

es o&&=130

494

Luis)Morellon)

Quantum)Hall)Effect))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


Klitzing)et)al.)1980)Klitzing,)Nobel)Prize)1985)

Si)MOSFET)2DEG,)T)=)4.2)K)HMFL)Grenoble)

Luis)Morellon)

Quantum)Hall)Effect))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


Filling)of)Landau)levels:))

σ xy = ne

2

h

RK =h

e2=µ0c

2α= 25,812.8074434(84) Ω

))))Klitzing’s)constant)__)>)metrology)(R)standard)))))))Accurate)to)1)in)109)))

<)1980:)states)of)maPer)__)>)spontaneous)symmetry)breaking))>)1980:)The)QHE)is)the)first)ordered)phase)that)has)no)spontaneous)symetry)breaking))The)behavior)does)not)depend)on)specific)geometry))THE)QHE)IS)TOPOLOGICALLY!DISTINCT!!

Luis)Morellon)

Topology))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


Topology:)It)is)the)study)of)geometrical)proper4es)that)are)preserved)under)con4nuous)deforma4ons)including)stretching)and)bending,)but)not)tearing)or)gluing.))Topological)invariant:)quan4ty)that)does)not)change)under)con4nuous)deforma4on.))Example:)2D)surfaces)in)3D))Genus)g)=)#)of)holes))Gaussian)curvature))κ)=)1)/)(R1R2)) g = 0 g = 1ral of the curvature o hole surface is “quantized”

.

from left to right, equators

κ)>)0)

κ)<)0)κ)=)0)

Luis)Morellon)

Topology))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


For)a)closed)surface:)Gauss)–)Bonnet)theorem)

∫dS κ = 2π(2 − 2g)

The)integral)of)the)curvature)over)the)surface)depends)only)on)global)proper4es)(topology))insensi4ve)to)small)changes)/deforma4on)of)surface)

geometry)<)__)__)>)topology)

g)is)an)integer)topological)invariant)

Series Editor:Professor Douglas F Brewer, MA, DPhil

Emeritus Professor of Experimental Physics, University of Sussex

GEOMETRY, TOPOLOGY

AND PHYSICS

SECOND EDITION

MIKIO NAKAHARA

Department of Physics

Kinki University, Osaka, Japan

INSTITUTE OF PHYSICS PUBLISHING

Bristol and Philadelphia

Luis)Morellon)

Berry)phase))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


Berry phase effects on electronic properties

Di Xiao

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,

Tennessee 37831, USA

Ming-Che Chang

Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan

Qian Niu

Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

REVIEWS OF MODERN PHYSICS, VOLUME 82, JULY–SEPTEMBER 2010

ψ(r) = eikruk(r)

A= i uk∇kuk

F =∇k× A

γ = A ⋅C∫ dk = dk ⋅F

S∫

Bloch)states)

Berry)connec4on)

Berry)curvature)

Berry)phase)

)(γ is)gauge)invariant))

Luis)Morellon)

TKKN))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


VOLUME 49, NUMBER 6 PHYSICAL REVIEW LETTERS 9 AUGUsr 1982

merized ground state for arbitrary values of M

and X (except M =0). The underlying reason isthat the phonon fluctuations induce an effectiveelectron-electron interaction of such a type thata CDW ground state is always produced. (Thatinteraction is ineffective in the case n = 1 forsmall coupling because of the Pauli excl.usion

principle ).This is accompanied by pairing of the

spin-up and spin-down electrons. However, this

conclusion is by no means inescapable. Prelim-inary numerical studies' show that other formsof the electron-phonon coupling (which induce

longer-range attraction) give a ground state with

superconducting correlations. This has also

been suggested from calculations based on per-turbation theory. ' The MC method used in this

paper offers the possibility of numerically study-

ing comp1. icated one-dimensional electron-phonon

models (the inclusion of electron-electron inter-action is straightforward) and thus investigating

the rich variety of ground-state phases for suchsystems, without restriction to a perturbativeregime.One of us (J.H. ) is indebted to D. Scalapino for

raising his interest in this problem and for nu-merous stimu1. ating discussions. We acknowledge

helpful conversations with S. Kivelson, W. P. Su,

R. Sugar, N. Andrei, S. Shenker, K. Maki,M. Stone, and particularly J. R. Schrieffer. One

of us (E.F.) thanks the Institute for TheoreticalPhysics for its kind hospitality during the summer

of 1981. This work was supported by the National

Science Foundation under Grants No. PHY77-

27084 and No. DMR81-17182.

'B. E. Peierls, Quantum Theory of Solids (Oxford,Univ. Press, London, 1955), p. 108.D. J. Scalapino and B. L. Sugar, Phys. Bev. B 24,

4295 (1981).3T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959).4W. P. Su, J.B. Schrieffer, and A. J. Heeger, Phys.

Bev. B 22, 2099 (1980).5J. E. Hirsch, D. J. Scalapino, B. L. Sugar, and

B. Blankenbecler, Phys. Bev. Lett. 47, 1628 (1981).6E. Fradkin and J.E. Hirsch, unpublished.

G. Beni, P. Pincus, and J. Kanamori, Phys. Bev.8 10, 1896 (1974).~W. P. Su, to be published.~J. E. Hirsch and D. J. Scalapino, unpublished.See V. J. Emery, in Kighly Conducting One-Dimen-

sional Sol'ids, edited by J. Devreese, B. Evrard, and

V. van Doren (Plenum, New York, 1979), and refer-ences therein.

Quantized Hall Conductance in a Two-Dimensional Periodic Potential

D. J. Thouless, M. Kohmoto, "'M. P. Nightingale, and M. den NijsDePa&ment of Physics, University of Washington, Seattle, Washington 98l95

(Received 30 April 1982)

The Hall conductance of a two-dimensional electron gas has been studied in a uniformmagnetic field and a periodic substrate potential U. The Kubo formula is written in aform that makes apparent the quantization when the Fermi energy lies in a gap. Explicitexpressions have been obtained for the Hall conductance for both large and small U/S~ .PACS numbers: 72.15.Gd, 72.20. Mg, 73.90.+b

The experimental discovery by von Klitzing,Dorda, and Pepper' of the quantization of the Hall

conductance of a two-dimensional electron gas ina strong magnetic field has led to a number of

theoretical studies of the problem. ' ' lt has beenconcluded that a noninteracting electron gas hasa Hall conductance which is a multiple of e'/h ifthe Fermi energy lies in a gap between Landau

levels, or even if there are tails of localizedstates from the adjacent Landau levels at the Fer-mi energy. However, it can be concluded from

Laughlin's' argument that the Hall conductance isquantized whenever the Fermi energy lies in anenergy gap, even if the gap lies within a Landaulevel. For example, it is known that if the elec-trons are subject to a weak sinusoidal perturba-tion as well as to the uniform magnetic field, with

p=p/q magnetic-flux quanta per unit cell of theperturbing potential, each Landau level is splitinto P subbands of equal weight.

'One might ex-

pect each of these subbands to give a Hall con-ductance equal to e'/ph, and that is what the clas-

1982 The American Physical Society 405

VOLUME 49, NUMBER 6 PHYSICAL REVIEW LETTERS 9 AUGUsr 1982

merized ground state for arbitrary values of M

and X (except M =0). The underlying reason isthat the phonon fluctuations induce an effectiveelectron-electron interaction of such a type thata CDW ground state is always produced. (Thatinteraction is ineffective in the case n = 1 forsmall coupling because of the Pauli excl.usion

principle ).This is accompanied by pairing of the

spin-up and spin-down electrons. However, this

conclusion is by no means inescapable. Prelim-inary numerical studies' show that other formsof the electron-phonon coupling (which induce

longer-range attraction) give a ground state with

superconducting correlations. This has also

been suggested from calculations based on per-turbation theory. ' The MC method used in this

paper offers the possibility of numerically study-

ing comp1. icated one-dimensional electron-phonon

models (the inclusion of electron-electron inter-action is straightforward) and thus investigating

the rich variety of ground-state phases for suchsystems, without restriction to a perturbativeregime.One of us (J.H. ) is indebted to D. Scalapino for

raising his interest in this problem and for nu-merous stimu1. ating discussions. We acknowledge

helpful conversations with S. Kivelson, W. P. Su,

R. Sugar, N. Andrei, S. Shenker, K. Maki,M. Stone, and particularly J. R. Schrieffer. One

of us (E.F.) thanks the Institute for TheoreticalPhysics for its kind hospitality during the summer

of 1981. This work was supported by the National

Science Foundation under Grants No. PHY77-

27084 and No. DMR81-17182.

'B. E. Peierls, Quantum Theory of Solids (Oxford,Univ. Press, London, 1955), p. 108.D. J. Scalapino and B. L. Sugar, Phys. Bev. B 24,

4295 (1981).3T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959).4W. P. Su, J.B. Schrieffer, and A. J. Heeger, Phys.

Bev. B 22, 2099 (1980).5J. E. Hirsch, D. J. Scalapino, B. L. Sugar, and

B. Blankenbecler, Phys. Bev. Lett. 47, 1628 (1981).6E. Fradkin and J.E. Hirsch, unpublished.

G. Beni, P. Pincus, and J. Kanamori, Phys. Bev.8 10, 1896 (1974).~W. P. Su, to be published.~J. E. Hirsch and D. J. Scalapino, unpublished.See V. J. Emery, in Kighly Conducting One-Dimen-

sional Sol'ids, edited by J. Devreese, B. Evrard, and

V. van Doren (Plenum, New York, 1979), and refer-ences therein.

Quantized Hall Conductance in a Two-Dimensional Periodic Potential

D. J. Thouless, M. Kohmoto, "'M. P. Nightingale, and M. den NijsDePa&ment of Physics, University of Washington, Seattle, Washington 98l95

(Received 30 April 1982)

The Hall conductance of a two-dimensional electron gas has been studied in a uniformmagnetic field and a periodic substrate potential U. The Kubo formula is written in aform that makes apparent the quantization when the Fermi energy lies in a gap. Explicitexpressions have been obtained for the Hall conductance for both large and small U/S~ .PACS numbers: 72.15.Gd, 72.20. Mg, 73.90.+b

The experimental discovery by von Klitzing,Dorda, and Pepper' of the quantization of the Hall

conductance of a two-dimensional electron gas ina strong magnetic field has led to a number of

theoretical studies of the problem. ' ' lt has beenconcluded that a noninteracting electron gas hasa Hall conductance which is a multiple of e'/h ifthe Fermi energy lies in a gap between Landau

levels, or even if there are tails of localizedstates from the adjacent Landau levels at the Fer-mi energy. However, it can be concluded from

Laughlin's' argument that the Hall conductance isquantized whenever the Fermi energy lies in anenergy gap, even if the gap lies within a Landaulevel. For example, it is known that if the elec-trons are subject to a weak sinusoidal perturba-tion as well as to the uniform magnetic field, with

p=p/q magnetic-flux quanta per unit cell of theperturbing potential, each Landau level is splitinto P subbands of equal weight.

'One might ex-

pect each of these subbands to give a Hall con-ductance equal to e'/ph, and that is what the clas-


σ xy = ne

2

h with n =

1

2πd

2kF

mBZ∫

m

∑ m)=)all)occupied)bands))(BZ)=)Brillouin)zone))

TKKN)topological)invariant))(first)Chern)number))(prePy)much)like)Gauss_Bonnet)theorem))

B

Insulating State

Quantum Hall State

E

k

0

E

k

EG

(a) (b) (c)

(d) (e) (f)

/a−π/a−π

0 /a−π/a−π

hωc

Hasan)&)Kane,)RMP)82,)3045)(2010))

Magnetic field

Quantum Hall system Quantum spin

Luis)Morellon)

Edge)states))))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


The!importance!of!the!edge!

momentum

energ

y

band gap edge states

conduction band

valence band

B

edge

“skipping)orbits”)

Halperin,)Phys.)Rev.)B)25,)2185)(1982))Büoker,)Phys.)Rev.)B)38,)9375)(1988))

•  chiral)gapless)edge)states)

•  n)=)#)of)edge)modes)

•  edge)modes)come)by)pair))(spin)symmetry))

•  no)backscaPering,)insensi4ve)to)disorder))

•  cannot)localize)

)

38 ABSENCE OF BACKSCATTERING IN THE QUANTUM HALL. . . 9377

II. THE TWO-TERMINAL CONDUCTANCE

A. Ideal perfect conductor

Consider an ideal two-dimensional conductor without

impurities or inhomogeneities of width w connecting two

electron reservoirs as shown in Fig. 2(a). The electron

reservoirs at chemical potentials p, and pz serve as

source and sink of carriers and of energy. A reservoir

emits carriers into current-carrying states up to its chem-

ical potential ~ Every carrier reaching a reservoir, in-

dependent of phase and of energy, is absorbed.Let us first brieAy consider the case of zero magnetic

field. The Hamiltonian of the perfect conductor is

(p„'+p,')+ v(y) .201

(2.1)

Here, x is the coordinate along the strip and y is the coor-

dinate transverse to the strip. The wave functions are se-

parable and of the form

Landauer approach which leads to Eq. (1.1) and thus to

resistances which are compatible with the (global)

Onsager-Casimir symmetry relations. An earlier ap-

proach defined resistance with regard to local electric

potentials in the perfect portions of the conductor away

from the terminals. There is no fundamental reason that

requires a resistance defined by invoking local electric po-

tentials to exhibit a particular symmetry. Indeed, as

shown in Ref. 10, this earlier formulation does not lead tothe reciprocity-symmetry equation (1.1). Experiments on

ultrasmall metallic lines, on macroscopic conductors ofvarious geometries,

' and on quantum Hall samples all

exhibit reciprocity symmetry. Therefore, it is clearly

necessary to use a formulation which leads to resistances

which are compatible with these fundamental sym-

metries.

A discussion of the quantum Hall effect, invoking Ref.

29 (local potentials away from terminals), has been given

by Streda et al. ' Jain and Kievelson ' 'apply the one-

channel Landauer formula. ' These papers study local

electric potentials in a two-terminal conductor. As in

Ref. 29, the piled-up charges are determined in the per-

fect portions of the conductor to the left and right of adisordered region only. The role of contacts is not ad-

dressed. In these papers' ' the longitudinal resistance

vanishes only if the two-terminal conductance is also

quantized. The authors of Refs. 19 and 20 obtain a "sum

rule" for the Hall resistance and the longitudinal resis-

tance, which is appropriate for three-terminal resis-

tances. ' In experiments, as pointed out already, the Hall

resistance and the longitudinal resistances are four-

terminal resistances. Four-terminal resistances obey a

more complex sum rule. ' Despite our criticism of thework of Streda et al. ,

'we emphasize its pioneering

character. Reference 19 has provided a large portion ofthe motivation for this paper. Beenakker and van

Houten and Peeters have pointed to the applicabilityof the four-terminal formula of Ref. 8 to the quantum

Hall effect and the work presented here proceeds in the

same direction.

FIG. 2. (a) Perfect two-dimensional conductor connected to

reservoirs. The chemical potentials of the reservoirs are p& and

p2. (b) Conductor with a disordered region (shaded part) con-

nected to the left and right to perfect conductors which, in turn,

connect to reservoirs.

I,=(elh)bp, (2.3)

independent of the channel index j. The total current is

I =N(e/h)b, p. Here, as in the remaining part of the pa-per, we assume that kT is small compared to the separa-tion of the transverse energy levels. The voltage drop be-

tween the reservoirs is eV=hp. Thus the two-terminal

resistance of a perfect ¹hannel wire is

1

ez N(2.4)

This result depends on the way current is fed from the

reservoir into the perfect conductor. Later, we shall con-

(2.2)

k is the wave vector along x, and f (y) is a transverseeigenfunction with energy eigenvalue E . The total ener-gy of the state is the sum of the transverse energy E and

the energy for longitudinal motion. Thus at the Fermi

energy EF=E +I k /2m, there are 2N states, where Nis the number of transverse energies E below the Fermi

energy. Let us calculate the current through this perfectconductor assuming p&&pz. Below pz left- and right-

moving states are equally occupied, and the net current is

zero. Thus we need to be concerned only with the energyinterval between pz and p, . The current injected by the

left reservoir in channel j is I =eu (dn/dE)jbp. Here,

uj=A' (dE//dk) is the longitudinal velocity at the Fer-

mi energy of channel j. (dn IdE), is the density of statesat the Fermi energy for this channel and bp=p,—pz. Inone dimension the density of states is dn /dk = I/2m. andhence (dnldE) = I/2Mu . Therefore, the current fed

into a channel is

9378 M. BUTTIKER 38

sider more realistic contacts and discuss how that

changes Eq. (2.4). For the remainder of this section we

continue to use this simple model of current-feeding and

current-drawing contacts.

Next, consider the perfect conductor in a magnetic

field. We take the vector potential A=(—By,0,0). TheHamiltonian is

0= p„— By—+p +V(y) . (2.5)

The magnetic field induces cyclotron motion of the car-

riers. The wave function is still separable and of the form

PJ ke'""f——~(y). This leads to an eigenvalue problem for

the function f,

3

o4~ ISIS/ 3LLI

2

I

J= 3

J=2

j= I

j=O

Ef= fi 8 m

, +—~,'(y&—y)'+ V(y) f . (2.6)2m Qy 2 Yo

Yp,

yo=—m co,

=—klan, (2.7)

where ls=(Pic/~ eB~

)' is the magnetic length. Con-

sider a range of y for which the confining potential is con-

stant. We take V(y) =—0. The solutions of Eq. (2.6) areharmonic-oscillator wave functions with a width propor-

tional to the magnetic length l~ with eigenvalues

E,„=A'a), (j+—,' ), (2.8)

where j=0, 1,2, . . . ; Eq. (2.8) is independent of the pa-

rameter yo (i.e., independent of k). This picture must

change near the edges of the sample at y& and y2. The

cyclotron motion is affected by the confining potential. '

Classically, the carriers perform motion along skipping

orbits. As a function of yo the eigenvalues depart from

the Landau formula, Eq. (2.8), and increase as the edge is

approached, as shown in Fig. 3. For a hard-wall poten-

tial, near the edges, the energy of a state depends on the

center yo through the distance y&—yo to the lower edge

and y2—yo to the upper edge. In general, the energy of astate is determined by '

E,k E(j,~„yo(——k)) . (2.9)

Using arguments similar to those applied to Bloch func-

tions, one can show that carriers in an edge state acquire

a longitudinal velocity,

dE,, dE,, dy,(2.10)

which is proportional to the slope of the Landau level.

dE/dyo is negative at the upper edge y2 and positive at

the lower edge y, (see Fig. 3). In a strong magnetic field

pointing out of the page, dyo/dk is negative and, there-

fore, the velocity along the upper edge is positive and

negative along the lower edge. Note that it is only the

edge states which can contribute to carrier flow becausethe bulk Landau states (the region in Fig. 3, where E is

««, ~, =~

eB~

/mc is the cyclotron frequency and m

the effective mass. In addition, the eigenvalues of Eq.(2.6) depend on the parameter

FIG. 3. Energy spectrum of a perfect conductor in a high

magnetic field for a rectangular confining potential (walls at y&and y&). The Landau levels at E, =%co,(j+ 2

j are strongly bent

upwards near the edges of the sample. yo is the center of theharmonic-oscillator wave functions. After Ref. 4.

independent of yo) have no velocity. The magnetic field

quenches the kinetic energy for longitudinal motion. The

density of states along a Landau level E can also be

found from dn /dk =- 1/2n appropriate for one-

dimensional conductors. Since dn /dk = (dn /dyo) ~

dyo/dk, we find, using Eq. (2.7), that

dn/dyo=2mlz. Away from the edges, the density ofstates is determined by a dense packing of cyclotron or-

bits in the plane. Further, the density of states is relatedto the velocity as in the conductor at zero field,

T

dn dn dk 1

dE . dk dE . 2m6v&k

The states at the Fermi energy are determined by the

equation EF Ejk, with E/——k given by Eq. (2.9). This

equation determines the values of k at the Fermi energy,k„(EF). There is a discrete number n=l, . . . ,N ofstates (N with positive k and N with negative k). As theFermi energy changes and passes through a bulk Landau

energy, the number of edge states intercepting the Fermi

energy drops discontinuously from N to N—l. The

current fed into each edge state is

dn eI=eu, (p,—p2)=—Ap .

. 1

Thus, the current fed into an edge state by a reservoir is

the same as the current fed into a quantum channel in azero-field perfect conductor. The resulting two-terminal

resistance for a perfect conductor in a high magnetic field

is thus

(2.11)

Here, N is the number of edge states (with positive veloci-

EF)n=3)

Landau)levels)

))Outline) ))

Luis)Morellon)



o !2D!TI’s!

o !3D!TI’s!



Luis)Morellon)

QSHE))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


In)QHE,)4me_reversal)symmetry)(T))is)broken))INGREDIENT:)SPIN_ORBIT)interac4on)

HSO

eff= λ(p×∇V ) ⋅S

•  analogous)to)magne4c)field)

•  opposite)force)for)opposite)spin)

•  energy)split)depends)on)spin)

•  does)not)break)T_symmetry)

•  Spin)Hall)Effect)(SHE))

SHE)))))))))))))))))))))))))))))))))))ISHE)

Fig. 1.1 Evolution from the ordinary Hall effect to the quantum spin Hall effect or two-

dimensional topological insulator. Here, B stands for a magnetic field, and M stands for

magnetization in a ferromagnet. The year means that the effect was discovered experimentally.

!H is the Hall conductance, and !S is the spin Hall conductance

2013!

VOLUME 61, NUMBER 18 PHYSICAL REVIEW LETTERS 31 OCTOBER 1988

Model for a Quantum Hall Eff'ect without Landau Levels:

Condensed-Matter Realization of the "Parity Anomaly"

F. D. M. Haldane

Department ofPhysics, University of California, San Diego, La Jolla, California 92093(Received 16 September 1987)

A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization

of the Hall conductance a" in the absence of an external magnetic field. Massless fermions without

spectral doubling occur at critical values of the model parameters, and exhibit the so-called "parity

anomaly" of (2+1)-dimensional field theories.

PACS numbers: 05.30.Fk, 11.30.Rd

The quantum Hall effect' (QHE) in two-dimensional

(2D) electron systems is usually associated with the pres-

ence of a uniform externally generated magnetic field,

which splits the spectrum of electron energy levels into

Landau levels. In this Letter I show how, in principle, a

QHE may also result from breaking of time-reversal

symmetry (i.e., magnetic ordering) without any net mag-

netic fiux through the unit cell of a periodic 2D system.

In this case, the electron states retain their usual Bloch

state character.

The model presented here is also interesting in that if

its parameters are on a critical line at which its ground

state changes from the normal semiconductor state to

this new type of QHE state, its low-energy states simu-

late a "(2+1)-dimensional" relativistic quantum field

theory exhibiting the so-called "parity anomaly" and a

(2+1)-D analog of "chiral" fermions without the

opposite-chirality anomaly-canceling partners that usu-

ally accompany them in lattice realizations of field

theories ("fermion doubling" ).In the zero-temperature limit, the transverse conduc-

tivity o "3' of a periodic 2D electron system with a gap in

the single-particle density of states at the Fermi level

takes quantized values ve /h, where v is generally ra-

tional, but can only take i nteger values in the absence of

electron interactions. This property of a pure system is

stable against sufficiently weak disorder effects. Sincea" is odd under time reversal, a nonzero value can only

occur if time-reversal invariance is broken.

In the usual QHE, the gap at the Fermi level results

from the splitting of the spectrum into Landau levels by

an external magnetic field. The scenario considered here

is different, and involves a 2D semimetal where there is a

degeneracy at isolated points in the Brillouin zone be-

tween the top of the valence band and the bottom of the

conduction band, that is associated with the presence ofboth inversion symmetry and time-reversal invariance.

If inversion symmetry is broken, a gap opens and the sys-

tem becomes a normal semiconductor (v=0), but if thegap opens because time-reversal invariance is broken the

system becomes a v=+ 1 integer QHE state. If bothperturbations are present, their relative strengths deter-

,bg qb, ~,

FIG. 1. The honeycomb-net model ("2D graphite") showing

nearest-neighbor bonds (solid lines) and second-neighbor bonds

(dashed lines). Open and solid points, respectively, mark the A

and 8 sublattice sites. The Wigner-Seitz unit cell is con-

veniently centered on the point of sixfold rotation symmetry

(marked "+")and is then bounded by the hexagon of nearest-

neighbor bonds. Arrows on second-neighbor bonds mark the

directions of positive phase hopping in the state with broken

time-reversal invariance.

mine which type of state is realized.

To model a 2D semimetal, I use the "2D graphite"

model investigated previously by Semenoff as a possible

lattice realization of a (2+I)-D field theory with the

anomaly. 2D graphite has the honeycomb net structure,

consisting of two interpenetrating triangular lattices("A" and "8"sublattices) with one lattice point of each

type per unit cell (Fig. 1). A 2D inversion (i.e., a rota-

tion in the plane by tr) interchanges the two sublattices.

Since spin-orbit coupling effects will not be included, the

electron spin will (for the moment) be suppressed.

Semenoff investigated the tight-binding model with

one orbital per site and a real hopping matrix element t ~

between nearest neighbors on different sublattices, and

also considered the effect of an inversion-symmetry-

breaking on-site energy +M on /I sites and —M on 8sites. The model has point group Cs„(M=O) or C3„(MAO). In this original version of the model, time-

reversal invariance is present, and Semenoff found com-

plete cancellation of the anomaly in the M =0 model dueto fermion doubling, and normal semiconductor behavior

for MAO.


VOLUME 61, NUMBER 18 PHYSICAL REVIEW LETTERS 31 OCTOBER 1988

Model for a Quantum Hall Eff'ect without Landau Levels:

Condensed-Matter Realization of the "Parity Anomaly"

F. D. M. Haldane

Department ofPhysics, University of California, San Diego, La Jolla, California 92093(Received 16 September 1987)

A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization

of the Hall conductance a" in the absence of an external magnetic field. Massless fermions without

spectral doubling occur at critical values of the model parameters, and exhibit the so-called "parity

anomaly" of (2+1)-dimensional field theories.

PACS numbers: 05.30.Fk, 11.30.Rd

The quantum Hall effect' (QHE) in two-dimensional

(2D) electron systems is usually associated with the pres-

ence of a uniform externally generated magnetic field,

which splits the spectrum of electron energy levels into

Landau levels. In this Letter I show how, in principle, a

QHE may also result from breaking of time-reversal

symmetry (i.e., magnetic ordering) without any net mag-

netic fiux through the unit cell of a periodic 2D system.

In this case, the electron states retain their usual Bloch

state character.

The model presented here is also interesting in that if

its parameters are on a critical line at which its ground

state changes from the normal semiconductor state to

this new type of QHE state, its low-energy states simu-

late a "(2+1)-dimensional" relativistic quantum field

theory exhibiting the so-called "parity anomaly" and a

(2+1)-D analog of "chiral" fermions without the

opposite-chirality anomaly-canceling partners that usu-

ally accompany them in lattice realizations of field

theories ("fermion doubling" ).In the zero-temperature limit, the transverse conduc-

tivity o "3' of a periodic 2D electron system with a gap in

the single-particle density of states at the Fermi level

takes quantized values ve /h, where v is generally ra-

tional, but can only take i nteger values in the absence of

electron interactions. This property of a pure system is

stable against sufficiently weak disorder effects. Sincea" is odd under time reversal, a nonzero value can only

occur if time-reversal invariance is broken.

In the usual QHE, the gap at the Fermi level results

from the splitting of the spectrum into Landau levels by

an external magnetic field. The scenario considered here

is different, and involves a 2D semimetal where there is a

degeneracy at isolated points in the Brillouin zone be-

tween the top of the valence band and the bottom of the

conduction band, that is associated with the presence ofboth inversion symmetry and time-reversal invariance.

If inversion symmetry is broken, a gap opens and the sys-

tem becomes a normal semiconductor (v=0), but if thegap opens because time-reversal invariance is broken the

system becomes a v=+ 1 integer QHE state. If bothperturbations are present, their relative strengths deter-

,bg qb, ~,

FIG. 1. The honeycomb-net model ("2D graphite") showing

nearest-neighbor bonds (solid lines) and second-neighbor bonds

(dashed lines). Open and solid points, respectively, mark the A

and 8 sublattice sites. The Wigner-Seitz unit cell is con-

veniently centered on the point of sixfold rotation symmetry

(marked "+")and is then bounded by the hexagon of nearest-

neighbor bonds. Arrows on second-neighbor bonds mark the

directions of positive phase hopping in the state with broken

time-reversal invariance.

mine which type of state is realized.

To model a 2D semimetal, I use the "2D graphite"

model investigated previously by Semenoff as a possible

lattice realization of a (2+I)-D field theory with the

anomaly. 2D graphite has the honeycomb net structure,

consisting of two interpenetrating triangular lattices("A" and "8"sublattices) with one lattice point of each

type per unit cell (Fig. 1). A 2D inversion (i.e., a rota-

tion in the plane by tr) interchanges the two sublattices.

Since spin-orbit coupling effects will not be included, the

electron spin will (for the moment) be suppressed.

Semenoff investigated the tight-binding model with

one orbital per site and a real hopping matrix element t ~

between nearest neighbors on different sublattices, and

also considered the effect of an inversion-symmetry-

breaking on-site energy +M on /I sites and —M on 8sites. The model has point group Cs„(M=O) or C3„(MAO). In this original version of the model, time-

reversal invariance is present, and Semenoff found com-

plete cancellation of the anomaly in the M =0 model dueto fermion doubling, and normal semiconductor behavior

for MAO.


Luis)Morellon)



-1

0

0 2π/aπ/a

E/t

k

1

X

X

Quantum Spin Hall Effect in Graphene

C. L. Kane and E. J. Mele

Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA(Received 29 November 2004; published 23 November 2005)

We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of

graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin

orbit potential converts graphene from an ideal two-dimensional semimetallic state to a quantum spin Hall

insulator. This novel electronic state of matter is gapped in the bulk and supports the transport of spin and

charge in gapless edge states that propagate at the sample boundaries. The edge states are nonchiral, but

they are insensitive to disorder because their directionality is correlated with spin. The spin and charge

conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba

coupling, disorder, and symmetry breaking fields are discussed.

PRL 95, 226801 (2005)P H Y S I C A L R E V I E W L E T T E R S week

25 NOVEMBER

The)edge)states)are)spin)filtered)

Luis)Morellon)



Z2 Topological Order and the Quantum Spin Hall Effect

C. L. Kane and E. J. Mele

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA(Received 22 June 2005; published 28 September 2005)

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic

band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is

associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2

classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number

classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band

model of graphene and propose a generalization of the formalism applicable to multiband and interacting

systems.

PRL 95, 146802 (2005)P H Y S I C A L R E V I E W L E T T E R S week ending

30 SEPTEMBER 2005

0 2π0 2π−1

0

1

−5 0 5

−5

0

5 I

QSH

λ / λR

λ / λv SO

SOE/t

ka kaπ π

(a) (b)

FIG. 1 (color online). Energy bands for a one-dimensional

‘‘zigzag’’ strip in the (a) QSH phase %v ! 0:1t and (b) the

insulating phase %v ! 0:4t. In both cases %SO ! :06t and %R !

:05t. The edge states on a given edge cross at ka ! &. The inset

shows the phase diagram as a function of %v and %R for 0<

%SO - t.

spin-down

spin-up

edge

momentum

energ

y

band gap edge states

momentum

conduction band

valence band

QSH)state)

2D)Bloch)Hamiltonian)T_symmetry) H (−k) =ΘH (k)Θ

−1

Θ2= −1

Z2)topological)invariant))))))))))))))(n)=)0;)TKKN))(ν)=)0,)1))

Luis)Morellon)

Formula)for)the)Z2)invariant))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


E

EF

Conduction Band

Valence Band

Quantum spin

Hall insulator ν=1

Conventional

Insulatorν=0

(a) (b)

k0/a−π /a−π

FIG. 5. !Color online" Edge states in the quantum spin Hall

insulator !QSHI". !a" The interface between a QSHI and an

ordinary insulator. !b" The edge state dispersion in the

graphene model in which up and down spins propagate in op-

posite directions.

um (k)

wmn (k) = um (k) Θ un (−k)

Θ2= −1⇒ w

T(k) = −w(−k)

wT(Λa ) = −w(Λa )

δ(Λa ) =Pf w(Λa )[ ]

det w(Λa )[ ]= ±1

(−1)ν = δ(Λa )a=1

4

∏ = ±1

•

•

•

•

4

1 2

3

kx

ky

Bulk 2D Brillouin Zone

•

•

•

•“time reversal polarization” analogous to

occupied)Bloch)states)

T_reversal)matrix)

an2symmetric)

fixed"point"parity)

Z2"invariant)

Luis)Morellon)

HgTe)QWs:)theory))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


QSH)insulator)=)2D)Topological)Insulator) Predicted:)Bernevig)et)al.)Science)314,)1757)(2006))Observed:)König)et)al.,)Science)318,)766)(2007))

)Quantum Spin Hall Effect andTopological Phase Transition inHgTe Quantum WellsB. Andrei Bernevig,1,2 Taylor L. Hughes,1 Shou-Cheng Zhang1*

We show that the quantum spin Hall (QSH) effect, a state of matter with topological propertiesdistinct from those of conventional insulators, can be realized in mercury telluride–cadmiumtelluride semiconductor quantum wells. When the thickness of the quantum well is varied, theelectronic state changes from a normal to an “inverted” type at a critical thickness dc. We show thatthis transition is a topological quantum phase transition between a conventional insulating phaseand a phase exhibiting the QSH effect with a single pair of helical edge states. We also discussmethods for experimental detection of the QSH effect.

www.sciencemag.org SCIENCE VOL 314 15 DECEMBER 2006Quantum Spin Hall Effect inTheory: Bernevig, Hughes and Zhang, Science ‘06

HgTe

HgxCd1-xTe

HgxCd1-xTed

d < 6.3 nm : Normal band order

Luis)Morellon)

HgTe)QWs:)theory))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


Theory: Bernevig, Hughes and Zhang, Science ‘06

d < 6.3 nm : Normal band order d > 6.3 nm : Inverted band order

Conventional InsulatorQuantum spin Hall Insulator

with topological edge states

6 ~ s

8 ~ p

k

E

6 ~ s

8 ~ p k

E

Egap~10 meV

2 ( ) 1n a 2 ( ) 1

n a

Band inversion transition:

Switch parity at k=0

lx-terminal

ofpin-uppin-

)-

uldh

onenae-

longitudinal−

haoisn)

e

X

X

4

2

µL

µ3

µ2µ1

µ4

µR

µFermi

Egap

Egap

µFermi

2

4

00

d < d c d > d cnormal regime inverted regime

GLRGLR

e2

h( )( )

e2

h

A

B

PREDICTION:!

G =2e

2

h

Luis)Morellon)

HgTe)QWs:)experiment))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)


Quantum Spin Hall Insulator Statein HgTe Quantum WellsMarkus König,1 Steffen Wiedmann,1 Christoph Brüne,1 Andreas Roth,1 Hartmut Buhmann,1

Laurens W. Molenkamp,1* Xiao-Liang Qi,2 Shou-Cheng Zhang2

Conductance

channel with

down-spin

charge carriers

Conductance

channel with

up-spin charge

carriers

Quantum

well

Schematic of the spin-polarized edge channels in a quantum spin Hallinsulator.

2 NOVEMBER 2007 VOL 318 SCIENCE766

–1.0 –0.5 0.0 0.5 1.0 1.5 2.010

3

104

105

106

107

R14,2

3 /

Ω

R14,2

3 / k

Ω

G = 0.3 e2/h

G = 0.01 e2/h

T = 30 mK

–1.0 –0.5 0.0 0.5 1.00

5

10

15

20

G = 2 e2/h

G = 2 e2/h

T = 0.03 K

(Vg – Vthr) / V

(Vg – Vthr) / V

T = 1.8 K

B)=)0);)T)=)30)mK)I:)5.5)nm)QW)normal)II:)7.3)nm)QW)inverted)III,)IV:)7.3)nm)QW,)L)=)1)µm))

Luis)Morellon)

TIs)


2010:!APS!March!MeePng!

s t -r n , e r h e -a r

t f l -n -g l

“There’s something about many-particle quantum mechanics that causes perfection to emerge out of imperfection.”

— Joel Moore

)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)

Luis)Morellon)

TIs)


Web!of!Science:!‘Topological!Insulators’!!!!!!!!!!!!!!!!!!!!!!!!!!(as!of!17/02/2015)!!

Published Items in Each Year

Citations in Each Year

)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)

))Outline) ))

Luis)Morellon)



o !2D!TI’s!

o !3D!TI’s!



k

kx

ky

(c) kz

Λ0,0,0

Λπ,0,0 Λ

π,π,0

Λ0,π,0

Λ0,π,π

Λπ,π,π

Λ0,0,π

Λπ,0,π

0 π

π

π

Luis)Morellon)

2D)!)3D)


)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)

Λi = Λ0, 0, 0, Λπ , 0, 0

, Λ0,π , 0

,

Λ0, 0, π , Λπ , 0, π , Λ

0, π , π ,

Λπ , π , 0, Λπ , π , π

δ(Λi ) =Pf w(Λi )[ ]

det w(Λi )[ ]

(−1)ν0 = δ(Λn1, n2 , n3

)n j=0, π

∏

(−1)νi = δ(Λn1, n2 , n3

)n j≠i=0, π

ni=π

∏

(i = 1, 2, 3)

In)3D)there)are)8)TRIMs,)leading)to)4)Z2)invariants:)(ν0;)ν1,)ν2,)ν3))(16)topological)classes))Edge)states)!)surface)states)(kx,)ky))

If)ν0)=)0))))“WEAK”)TI)layered)QSHI)where)(ν1,)ν2,)ν3))=)(h"k"l"))direc4on)of)layers)Unlike)2D_QSHI,)T_symmetry)does)not)protect)the)surface)states)Fermi)surface)encloses)an)EVEN)no.)of)Dirac)points)

Fu,)Kane,)Mele,)PRL)98)(2007))106803)Moore)&)Balents)PRB)75)(2007))121306(R))Ray,)PRB)79)(2009))195322)

(a)

kx

ky

Γ4

Γ3

Γ1

Γ2

Luis)Morellon)

2D)!)3D)



4)Z2)invariants:))(ν0;)ν1,)ν2,)ν3))

If)ν0)=)1))))“STRONG”)TI)less)obvious)Fermi)surface)encloses)an)ODD)no.)of)Dirac)points)(1?))Robust)to)disorder)

)(b) (c)

EF

Ekx

ky

Γ4

Γ3

Γ1

Γ2

PROPOSAL:)Bi1_xSbx)Again)band)inversion!!) Bi1-xSbx

0

8

( 1) ( )n i

Inversion symmetry

EF

Pure Bismuthsemimetal

Alloy : .09<x<.18semiconductor Egap ~ 30 meV

Pure Antimonysemimetal

Ls

La Ls

La

Ls

LaEF EF

Egap

T L T L T L

E

k

Luis)Morellon)

STRONG)TI:)Bi1_xSbx)



Predict Bi Sb is a strong topological insulator: (1 ; 111).

1i n

Inversion symmetry:

(−1)ν0 = ξ2n (Γi

)n

∏i=1

8

∏

ξ2n (Γi) = ±1 parity of band 2n at Γ

i

HOW)TO)PROBE)EXPERIMENTALLY?))

•  ARPES)•  TRANSPORT)MEASUREMENTS)•  STM)/)STS)

PROPOSAL:)Bi1_xSbx)Again)band)inversion!!)

Luis)Morellon)

ARPES)



ARPES)=)Angle)Resolved)Photoemission)Spectroscopy)•  Directly)surface)electron)dispersion)•  Spin)texture)

of

)

of

t

,

,

of

-

A C

h

e

y

x

z

x’y’

z’

sample

Luis)Morellon)

ARPES)



See)e.g.)Shen)Lab)at)hPp://arpes.stanford.edu/index.html)

Luis)Morellon)

BiSb)experiment)



LETTERS

A topological Dirac insulator in a quantum spin HallphaseD. Hsieh1, D. Qian1, L. Wray1, Y. Xia1, Y. S. Hor2, R. J. Cava2 & M. Z. Hasan1,3

Vol 452 |24 April 2008 |doi:10.1038/nature06843

The)first)3D)TI:)Bi0.9Sb0.1)(1)1)1))single)xtal)

0 100 200 3000

2

4

6

8

! 80

"(m

#cm

)

T (K)

x=0x=0.1

1 2

0.0 0.2 0.4 0.6 0.8 1.0

$ Mk (Å )X

-1

0.0

-0.1

E(e

V)

B

0.1

3 4 5,

-(KP) (KP)

bulkgap

(a)

3D Topo. Insulator (Bi Sb1-x x)

(c)

T

K$

k

z

1

L

2L

X

x

M

X

ky

E

x

T

L

L

S

a

4% 7% 8%Bi

(b)

Γa Γb

Valence Band

Conduction Band

FE

kk

(b)E

Surface)states)cross)the)Fermi)level)5)4mes)(odd))

Surface)states)topologically)protected)

Bi0.9Sb0.1)is)a)strong)3D)TI)with)ν0)=)1)

Topological)class)(1;)1,)1,)1))

)

Luis)Morellon)

2nd)genera4on:)Bi2Se3,)Bi2Te3,)Sb2Te3)



Search)for)Tis)with)larger)band)gap)and)simpler)surface)spectrum)Xia)et)al.)Nat.)Phys.)5,)398)(2009))Zhang)et)al.,)Nat.)Phys.)5,)438)(2009)))

Luis)Morellon)

2nd)genera4on:)Bi2Se3,)Bi2Te3,)Sb2Te3)



Experimental Realization of aThree-Dimensional TopologicalInsulator, Bi2Te3Y. L. Chen,1,2,3 J. G. Analytis,1,2 J.-H. Chu,1,2 Z. K. Liu,1,2 S.-K. Mo,2,3 X. L. Qi,1,2 H. J. Zhang,4

D. H. Lu,1 X. Dai,4 Z. Fang,4 S. C. Zhang,1,2 I. R. Fisher,1,2 Z. Hussain,3 Z.-X. Shen1,2*

REPORTS10 JULY 2009 VOL 325 SCIENCE www.sciencemag.org178

(b)(a)

(b)

kk = 0

Bi2Te3

En

erg

y

Bi2Se3

En

erg

y

kk = 0

ky

Γ

spin

vF

k1

kx

(d)ky

Γ

spin

vF

k1

kx

(c)

Fig. 8. (Color online) Schematic bulk and surface band structures of

(a) Bi2Se3 and (b) Bi2Te3. Note that the surface states are spin non-

degenerate and are helically spin polarized. Representative constant-energy

contours of the Dirac cones for (c) Bi2Se3 and (d) Bi2Te3 are also

schematically shown. Note that the spin vector is always perpendicular to

the wave vector k1 in both Bi2Se3 and Bi2Te3, but the Fermi velocity vector

vF can be non-orthogonal to the spin vector in Bi2Te3 due to the hexagonal

warping, which leads to strong quasiparticle interference.

Luis)Morellon)

Transport)proper4es)



Signatures)in)transport)of)the)2D)topological)surface)states:))•  WAL)•  SdH)oscilla4ons)

PosiPve!MR!

Luis)Morellon)

Transport)proper4es)



Weak Anti-localization and QuantumOscillations of Surface States inTopological Insulator Bi2Se2TeLihong Bao1

*, Liang He2*, Nicholas Meyer1, Xufeng Kou2, Peng Zhang3, Zhi-gang Chen4, Alexei V. Fedorov3,Jin Zou4, Trevor M. Riedemann5, Thomas A. Lograsso5, Kang L. Wang2, Gary Tuttle1 & Faxian Xiu1

A

Received6 September 2012

Accepted25 September 2012

Published11 October 2012

SCIENTIFIC

Lihong Bao1 , Liang He2 , Nicholas Meyer1, Xufeng Kou2, Peng Zhang3, Zhi-gang Chen4, Alexei V. Fedorov3,Jin Zou4, Trevor M. Riedemann5, Thomas A. Lograsso5, Kang L. Wang2, Gary Tuttle1 & Faxian Xiu1

A




SCIENTIFIC

www.nature.com/

SCIENTIFIC

www.nature.com/

SCIENTIFIC

tw=tSO and tw=te, the conduction correction is

:G(B)G(0)%ae2

2p2Y(

1

2z

Bw

B) ln (

Bw

B)

! "

,

sing time, t (t ) is spin-orbit (elastic) scattering

www.nature.com/

SCIENTIFIC

by dGWAL(B)

www.nature.com/

SCIENTIFIC

Hikami_Larkin_Nagaoka)

www.nature.com/

SCIENTIFIC

α)≈)_)1/2))))))Lφ)≈)T_)1/2))

2D)transport)

))Outline) ))

Luis)Morellon)



o !2D!TI’s!

o !3D!TI’s!



Luis)Morellon)

STM)of)TIs)



l

cture

trong

one

aker,

tions,

Bi

Te

Se

Quintuple

layers

Single crystal provided

Single)crystal)of)Bi2Se2Te)provided)by))T.)Lograsso’s)group,)Ames)Lab.)(001) surface.

Weak Anti-localization and QuantumOscillations of Surface States inTopological Insulator Bi2Se2TeLihong Bao1

*, Liang He2*, Nicholas Meyer1, Xufeng Kou2, Peng Zhang3, Zhi-gang Chen4, Alexei V. Fedorov3,Jin Zou4, Trevor M. Riedemann5, Thomas A. Lograsso5, Kang L. Wang2, Gary Tuttle1 & Faxian Xiu1

A




SCIENTIFIC


A




SCIENTIFIC


A




SCIENTIFIC REPORTS | 2 : 726 | DOI: 10.1038/srep00726

www.nature.com/

SCIENTIFIC

ARPES))

!

momentum.

s

Moncayo: JT-STM by SPECS GmbH

• T=1.1 K

• W tip

• Spectroscopy modulation 2-5 mV rms

• P≤10-10 mbar

• Axial magnetic field 3 Tesla

Experimental

dI/d

V (

a.u

)

Luis)Morellon)

STM)of)TIs)



STM:)atomic)resolu4on)

Collabora4on)with)D.)Serrate)PhD)thesis)of)M.)C.)Marnez_Velarte))

Luis)Morellon)

STM)of)TIs)



STS:)dI/dV)≈)LDOS)(EF)+)eV))•  Features)in)LDOS)

•  Energy)gap)of) 700)meV)

•  The)FFT)shows)no)clear)features.)

Ag (111) surface states

450 mVq"resolu4on"q)≤)0.03)Å_1)

!600 !400 !200 0 200 400 600

VB C B

)

)

dI/dV)(a.u)

Vbias

)(mV )

S urfa ce )

S ta tes

Control)experiment)

Luis)Morellon)

STM)of)TIs)



Co)atoms)evapora4on)

Vbias)=)_500)mV,)I)=)50pA))Vbias)=))2)V,)I)=)50pA)

Pris4ne)surface) Co)atoms)

Luis)Morellon)

New)TIs)


)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New!TIs)))))))))))))))Conclusions)

Table I. Summary of topological insulator materials that have bee experimentally addressed. The definition of (1;111) etc. is introduced in Sect. 3.7.

(In this table, S.S., P.T., and SM stand for surface state, phase transition, and semimetal, respectively.)

Type Material Band gap Bulk transport Remark Reference

2D, ! ¼ 1 CdTe/HgTe/CdTe <10meV insulating high mobility 31

2D, ! ¼ 1 AlSb/InAs/GaSb/AlSb #4meV weakly insulating gap is too small 73

3D (1;111) Bi1!xSbx <30meV weakly insulating complex S.S. 36, 40

3D (1;111) Sb semimetal metallic complex S.S. 39

3D (1;000) Bi2Se3 0.3 eV metallic simple S.S. 94

3D (1;000) Bi2Te3 0.17 eV metallic distorted S.S. 95, 96

3D (1;000) Sb2Te3 0.3 eV metallic heavily p-type 97

3D (1;000) Bi2Te2Se #0:2 eV reasonably insulating "xx up to 6! cm 102, 103, 105

3D (1;000) (Bi,Sb)2Te3 <0:2 eV moderately insulating mostly thin films 193

3D (1;000) Bi2!xSbxTe3!ySey <0:3 eV reasonably insulating Dirac-cone engineering 107, 108, 212

3D (1;000) Bi2Te1:6S1:4 0.2 eV metallic n-type 210

3D (1;000) Bi1:1Sb0:9Te2S 0.2 eV moderately insulating "xx up to 0.1! cm 210

3D (1;000) Sb2Te2Se ? metallic heavily p-type 102

3D (1;000) Bi2(Te,Se)2(Se,S) 0.3 eV semi-metallic natural Kawazulite 211

3D (1;000) TlBiSe2 #0:35 eV metallic simple S.S., large gap 110–112

3D (1;000) TlBiTe2 #0:2 eV metallic distorted S.S. 112

3D (1;000) TlBi(S,Se)2 <0:35 eV metallic topological P.T. 116, 117

3D (1;000) PbBi2Te4 #0:2 eV metallic S.S. nearly parabolic 121, 124

3D (1;000) PbSb2Te4 ? metallic p-type 121

3D (1;000) GeBi2Te4 0.18 eV metallic n-type 102, 119, 120

3D (1;000) PbBi4Te7 0.2 eV metallic heavily n-type 125

3D (1;000) GeBi4!xSbxTe7 0.1–0.2 eV metallic n (p) type at x ¼ 0 (1) 126

3D (1;000) (PbSe)5(Bi2Se3)6 0.5 eV metallic natural heterostructure 130

3D (1;000) (Bi2)(Bi2Se2:6S0:4) semimetal metallic (Bi2)n(Bi2Se3)m series 127

3D (1;000) (Bi2)(Bi2Te3)2 ? ? no data published yet 128

3D TCI SnTe 0.3 eV (4.2K) metallic Mirror TCI, nM ¼ !2 62

3D TCI Pb1!xSnxTe <0:3 eV metallic Mirror TCI, nM ¼ !2 164

3D TCI Pb0:77Sn0:23Se invert with T metallic Mirror TCI, nM ¼ !2 162

2D, ! ¼ 1? Bi bilayer #0:1 eV ? not stable by itself 82, 83

3D (1;000)? Ag2Te ? metallic famous for linear MR 134, 135

3D (1;111)? SmB6 20meV insulating possible Kondo TI 140–143

3D (0;001)? Bi14Rh3I9 0.27 eV metallic possible weak 3D TI 145

3D (1;000)? RBiPt (R = Lu, Dy, Gd) zero gap metallic evidence negative 152

Weyl SM? Nd2(Ir1!xRhx)2O7 zero gap metallic too preliminary 158

Luis)Morellon)

New)TIs)



The Complete Quantum Hall Trio

PHYSICS

Seongshik Oh

Observation of a quantized resistance state

in the absence of an external magnetic fi eld

completes a trio of quantum Hall related effects.

Quantum Hall

(1980)

H M

Quantum Hall

Quantum spin Hall(2007)

Quantum spin Hall Quantum anomalous Hall

Quantum anomalous Hall

(2013)

Hall

(1879)

Spin Hall

(2004)

Anomalous Hall

(1881) PREDICTION:!

σ xy =e2

h

9. R. Yu et al., Science 329, 61 (2010).

Experimental Observation of theQuantum Anomalous Hall Effectin a Magnetic Topological InsulatorCui-Zu Chang,1,2* Jinsong Zhang,1* Xiao Feng,1,2* Jie Shen,2* Zuocheng Zhang,1 Minghua Guo,1

Kang Li,2 Yunbo Ou,2 Pang Wei,2 Li-Li Wang,2 Zhong-Qing Ji,2 Yang Feng,1 Shuaihua Ji,1

Xi Chen,1 Jinfeng Jia,1 Xi Dai,2 Zhong Fang,2 Shou-Cheng Zhang,3 Ke He,2† Yayu Wang,1† Li Lu,2

Xu-Cun Ma,2 Qi-Kun Xue1†

www.sciencemag.org SCIENCE VOL 340 12 APRIL 2013

EXPERIMENT:!

Luis)Morellon)

New)TIs)



30 mK

30 mK

A B

C D

www.sciencemag.org SCIENCE VOL 340 12 APRIL 2013

Luis)Morellon)

New)TIs)



Okay,!I!understood!!!Then,!can!we!observe!a!QHE!coming!from!the!!

topological!surface!states!in!a!3D!TI?!!

ARTICLESPUBLISHED ONLINE: 10 NOVEMBER 2014 | DOI: 10.1038/NPHYS3140

Observation of topological surface state quantumHall eect in an intrinsic three-dimensionaltopological insulator

Yang Xu1,2, Ireneusz Miotkowski1, Chang Liu3,4, Jifa Tian1,2, Hyoungdo Nam5, Nasser Alidoust3,4,

Jiuning Hu2,6, Chih-Kang Shih5, M. Zahid Hasan3,4 and Yong P. Chen1,2,6*

Oh!man,!you’re!already!late!!)

Luis)Morellon)

Not)covered)today)


)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions!

•  Axion)electrodynamics)

•  Magnetoelectric)response)

•  Majorana)fermions)

•  Quantum)computa4on)

•  Energy)applica4ons)

•  New)TI)materials)

•  …)

Thank!you!for!your!a_enPon!!

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