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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)
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)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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-
1982 The American Physical Society 405
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)
Topological)Insulators)
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)
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)
Luis)Morellon)
QSHE))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
Topological)Insulators)
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.
1988 The American Physical Society 2015
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.
1988 The American Physical Society 2015
Luis)Morellon)
QSHE))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
Topological)Insulators)
-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)
QSHE))))))))))QHE )))))))))))))))))))2D!TIs!!!!!!!!!!!!!!!!!!3D)TIs)))))))))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
Topological)Insulators)
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)
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)
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)
Topological)Insulators)
)))))))))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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
See)e.g.)Shen)Lab)at)hPp://arpes.stanford.edu/index.html)
Luis)Morellon)
BiSb)experiment)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
Signatures)in)transport)of)the)2D)topological)surface)states:))• WAL)• SdH)oscilla4ons)
PosiPve!MR!
Luis)Morellon)
Transport)proper4es)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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
Received6 September 2012
Accepted25 September 2012
Published11 October 2012
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)
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)
Luis)Morellon)
STM)of)TIs)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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
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
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
Received6 September 2012
Accepted25 September 2012
Published11 October 2012
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
STM:)atomic)resolu4on)
Collabora4on)with)D.)Serrate)PhD)thesis)of)M.)C.)Marnez_Velarte))
Luis)Morellon)
STM)of)TIs)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D!TIs!!!!!!!))))))))))))))))))))))New)TIs)))))))))))))))Conclusions)
Co)atoms)evapora4on)
Vbias)=)_500)mV,)I)=)50pA))Vbias)=))2)V,)I)=)50pA)
Pris4ne)surface) Co)atoms)
Luis)Morellon)
New)TIs)
Topological)Insulators)
)))))))))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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New!TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New!TIs)))))))))))))))Conclusions)
30 mK
30 mK
A B
C D
www.sciencemag.org SCIENCE VOL 340 12 APRIL 2013
Luis)Morellon)
New)TIs)
Topological)Insulators)
)))))))))QHE )))))))))))))))))))2D)TIs))))))))))))))))))3D)TIs)))))))))))))))))))))))))))))New!TIs)))))))))))))))Conclusions)
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)
Topological)Insulators)
)))))))))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!!