Topological insulators and superconductors
KITPC 2010
Shoucheng Zhang, Stanford University
Colloborators
Stanford group: Xiaoliang Qi, Andrei Bernevig, Congjun Wu, Chaoxing Liu,
Taylor Hughes, Sri Raghu, Suk-bum Chung
Stanford experimentalists: Yulin Chen, Ian Fisher, ZX Shen, Yi Cui, Aharon
Kapitulnik, …
Wuerzburg colleagues: Laurens Molenkamp, Hartmut Buhmann, Markus Koenig,
Ewelina Hankiewicz, Bjoern Trauzettle
IOP colleagues: Zhong Fang, Xi Dai, Haijun Zhang, …
Tsinghua colleagues: Qikun Xue, Jinfeng Jia, Xi Chen,…
Outline
Models and materials of topological insulators
General theory of topological insulators, exotic particles
Topological superconductors
The search for new states of matter
The search for new elements led to a golden age of chemistry.
The search for new particles led to the golden age of particle physics.
In condensed matter physics, we ask what are the fundamental states of matter?
In the classical world we have solid, liquid and gas. The same H2O molecules can
condense into ice, water or vapor.
In the quantum world we have metals, insulators, superconductors, magnets etc.
Most of these states are differentiated by the broken symmetry.
Crystal: Broken
translational symmetry
Magnet: Broken
rotational symmetry
Superconductor: Broken
gauge symmetry
The quantum Hall state, a topologically non-trivial
state of matter
 xy
e2
n
h
• TKNN integer=the first Chern number.
d 2 k 
n
 F (k )
2
(2 )
• Topological states of matter are
defined and described by topological
field theory:
S eff 
 xy
2
2

d
xdt

A  A

• Physically measurable topological
properties are all contained in the
topological field theory, e.g. QHE,
fractional charge, fractional statistics
etc…
• von Klitzing, 1980
Discovery of the 2D and 3D topological insulator
HgTe Theory: Bernevig, Hughes and Zhang, Science 314, 1757 (2006)
Experiment: Koenig et al, Science 318, 766 (2007)
BiSb Theory: Fu and Kane, PRB 76, 045302 (2007)
Experiment: Hsieh et al, Nature 452, 907 (2008)
Bi2Te3, Sb2Te3, Bi2Se3 Theory: Zhang et al, Nature Physics 5, 438 (2009)
Experiment Bi2Se3: Xia et al, Nature Physics 5, 398 (2009),
Experiment BieTe3: Chen et al Science 325, 178 (2009)
On average 2-3 paper per day on the subject!
Topological Insulator is a New State of Quantum Matter
From traffic jam to info-superhighway
on chip
Traffic jam inside chips today
Info highways for the chips in the future
Quantum Hall effect and quantum spin Hall effect
Topological protection
(Qi and Zhang, Phys Today, Jan, 2010)
Spin=1/2
y>-y
The topological distinction between a conventional
insulator and a QSH insulator
Kane and Mele, Wu, Bernevig and Zhang; Xu and Moore
• Band diagram of a conventional insulator, a conventional insulator with accidental
surface states (with animation), a QSH insulator (with animation). Blue and red
color code for up and down spins.

k
Trivial
k=0 or 
Trivial
Non-trivial
Bandgap vs. lattice constant
(at room temperature in zinc blende structure)
6.0
5.5
5.0
Bandgap energy (eV)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
lattice constant a 0 [Å]
6.2
6.3
6.4
6.5
6.6
6.7
Band Structure of HgTe
S
P
P3/2
S
S
S
P1/2
P
P3/2
P1/2
Band inversion in HgTe leads to a topological quantum
phase transition
Let us focus on E1, H1
bands close to
crossing point
HgTe
HgTe
H1
E1
CdTe
H1
normal
CdTe
CdTe
CdTe
E1
inverted
The model of the 2D topological insulator
(BHZ, Science 2006)
Square lattice with 4-orbitals per site:
s,  , s,  , ( p x  ip y ,  ,  ( p x  ip y ), 
Nearest neighbor hopping integrals. Mixing matrix
elements between the s and the p states must be
odd in k.
0 
 h(k )

H eff (k x , k y )  

h ( k ) 
 0
m(k )
A(sin k x  i sin k y ) 

  d a (k ) a
h(k )  

A
(sin
k

i
sin
k
)

m
(
k
)
x
y


 m  Bk 2
A(k x  ik y ) 

 
2 
 A(k x  ik y )  m  Bk 
Similar to relativistic Dirac equation in 2+1 dimensions, with a mass term
tunable by the sample thickness d! m/B<0 for d>dc.
Mass domain wall
Cutting the Hall bar along the y-direction we see a domain-wall structure in the
band structure mass term. This leads to states localized on the domain wall
which still disperse along the x-direction, similar to Jackiw-Rebbi soliton.
y
y
x
m/B<0
0
x
E
E
kx
Bulk
0
Bulk
m
Experimental setup
• High mobility samples
of HgTe/CdTe quantum
wells have been
fabricated.
• Because of the small
band gap, about several
meV, one can gate dope
this system from n to p
doped regimes.
• Two tuning parameters,
the thickness d of the
quantum well, and the
gate voltage.
•(Koenig et al, Science 2007)
Experimental observation of the QSH edge state
(Konig et al, Science 2007)
x
x
Nonlocal transport in the QSH regime, (Roth et al Science
2009)
25
20
R (k)
R14,14=3/4 h/e2
15
1
2
I: 1-4
10
V: 2-3
4
5
3
R14,23=1/4 h/e2
0
0.0
0.5
1.0
V* (V)
1.5
2.0
No QSH in graphene
•
•
•
•
•
Bond current model on honeycomb
lattice (Haldane, PRL 1988).
Spin Hall insulator (Murakami,
Nagaosa and Zhang 2004)
Spin-orbit coupling in graphene
(Kane and Mele 2005). They took
atomic spin-orbit coupling of 5meV
to estimate the size of the gap.
Yao et al, Min et al 2006 showed
that the actual spin-orbit gap is
given by D2so/D =10-3 meV.
Similar to the seasaw mechanism of
the neutrino mass in the Standard
Model!
Single valley 2D Dirac cone
•
•
•
At the critical thickness, the
BHZ model reduces to the
massless Dirac model in
2+1d, with a single valley.
Recent experiments on HgTe
has reached this critical
point!
HgTe provides an ideal
platform to study 2D
massless Dirac fermions!
3D insulators with a single Dirac cone on the surface
z
(a)
(b)
y
y
x
Quintuple
layer
x
(c)
C
t1
t2
t3
Bi
Se1
Se2
A
B
C
A
B
C
Relevant orbitals of Bi2Se3 and the band inversion
(a)
(b)
0.6
E (eV)
Bi
Se
0.2
c
-0.2
0
(I)
(II)
(III)

0.2
(eV)
0.4
(a) Sb2Se3
(b) Sb2Te3
(c) Bi2Se3
(d) Bi2Te3
Model for topological insulator Bi2Te3, (Zhang et al, 2009)
Pz+, up, Pz-, up, Pz+, down, Pz-, down
Single Dirac cone on the surface of Bi2Te3
Surface of Bi2Te3 = ¼ Graphene !
Arpes experiment on Bi2Te3 surface states, Shen group
Doping evolution of the FS and band structure
Undoped
Under-doped
Optimallydoped
Over-doped
EF(undoped)
BCB bottom
BVB bottom
Dirac point position
Arpes experiment on Bi2Se3 surface states, Hasan group
3D TI
Surface states of TIs vs Rashba SOC
 Unlike graphene, here two components are related by time
reversal, and Pauli matrix  is the real spin.
 Surface Rashba term: Breaking of inversion symmetry at
the surface.
Conduction band
Conduction band
E
Strong
SOC
Valence band
E
Valence band
Spin-plasmon collective mode (Raghu, Chung, Qi+SCZ, PRL2009)
• General operator identity:
• Density-spin coupling:
Surface Landau levels, Xue group
General theory of topological insulators
• Topological field theory of
topological insulators. Generally
valid for interacting and
disordered systems. Directly
measurable physically. Relates to
axion physics! (Qi, Hughes and
Zhang)
• For a periodic system, the system is time
reversal symmetric only when
q=0 => trivial insulator
q= => non-trivial insulator
• Topological band theory based
on Z2 topological band invariant
of single particle states.
(Fu, Kane and Mele, Moore and Balents,
Roy)
Generalization of the QH topology state in d=2 to
time reversal invariant topological state in d>2, in
Science 2001
The periodic table of topological states:
(Qi, Hughes and Zhang, Ludwig et al, Kitaev)
The TRI topological insulators form a dimensional chain:
4D=>3D=>2D
The 4D TRI topological insulator directly inspired the
discovery of the intrinsic spin Hall effect in d=3,
Science 2003
“Recently, the QHE has been generalized to four spatial dimensions (4). In that case, an
electric field induces an SU(2) spin current through the nondissipative transport
equation…The quantum Hall response in that system is physically realized though the spinorbit coupling in a time reversal symmetric system.”
TRB topological insulators in d=2
• Chern-Simons topological field theory, odd under TR:
• 1st Chern number
TRI topological insulators in d=4
• Chern-Simons topological field theory, even under TR:
• If we perform Kaluza-Klein compactification, we obtain the effective field
theory of the d=3 topological insulator!
• 2st Chern number
• If we replace k4 by an adiabatic parameter, we obtain the quantized cyclic
change of the magneto-electric polarization in d=3!
The best way to understand TRI TI is D=4 => D=3
=> D=2
The best way to understand critical phenomenon is
D=4 => D=4-
Dimensional reduction
• From 4D QHE to the 3D topological insulator
Zhang & Hu, Qi, Hughes & Zhang
S4 DQH   d 4 xdt  A F F
  d xdt(  dx5 A5 ( x, t ))
3
 S3D   d xdtq ( x, t )
3


A5
x5
F F
q
(x, y, z)
F F
• From 3D axion action to the 2D QSH
S3D   d 3 xdt  A  q  A
  d 2 xdt  (  dzAz ( x, t ))  q  A
 S2 D   d 2 xdt     q A
e 
J 2 D  2      q
2
e 

 J 1D     
2

Goldstone & Wilzcek
TRI topological insulators in d=3
• Axion field theory, even under TR only when q=0, :
• Magneto-electric polarization (QHZ 2008):
Generalization to general interacting TI
• Topological order parameter
for generally interacting TI
• Experimentally measurable
through the topological
magneto-electric effect
• WZW extension u introduces
integer ambiguity of P3
• P3 is topologically quantized to
be integer or half-integer
• Also applies to disordered
systems, see Li et al, Groth et al.
(Wang, Qi, SCZ)
Topological field theory and the family tree
• Topological field theory of the QHE:
(Thouless et al, Zhang, Hansson and Kivelson)
S   d 2 k da (k )  d 3 x A( x)  dA( x)
• Topological field theory of the TI:
(Qi, Hughes and Zhang, 2008)
S   d 3 k (a (k )  da (k )  ..) d 4 x dA( x)  dA( x)
• More extensive and
general classification
soon followed (Kitaev,
Ludwig et al)
TRI topological insulator d=2, characterized by the
discrete Z2 topological number in 2005
Topological band theory
The continuum TI formula and the discrete TI formula
are pre-destined to converge! (Wang, Qi and Zhang, 0910.5954)
Equivalence between the integral and the discrete
topological invariants (Wang, Qi and Zhang, 0910.5954)
LHS=QHZ definition of TI, RHS=FKM definition of TI
RKKY coupling of the surface states
(Liu et al, PRL 2009)
Low frequency Faraday/Kerr rotation
(Qi, Hughes and Zhang, PRB78, 195424, 2008, Zhang group 2010, MacDonald group 2010)
Adiabatic
Requirement:
(surface gap)
Universal
quantization in units
of the fine structure
constant!
Seeing the magnetic monopole thru the mirror of a TME
insulator, (Qi et al, Science 323, 1184, 2009)
q
TME insulator
(for =’, =’)
higher order
feed back
similar to Witten’s dyon effect
Magnitude of B:
Dynamic axions in topological magnetic insulators
(Li et al, Nature Physics 2010)
• Hubbard interactions leads to anti-ferromagnetic order
• Effective action for dynamical axion
Axions and dark matter on your desktop?
(Zhang group, Nature
Physics 2010)
Now Shou-Cheng Zhang and his
colleagues inform us that, all along,
axions have been lurking
unrecognized on surfaces
of topological insulators.
Frank Wilzcek, NATURE 458, 129
(2009)
Topological insulators and superconductors
Full pairing gap in the bulk, gapless Majorana edge and surface states
Chiral Majorana fermions
Chiral fermions
massless Majorana fermions
massless Dirac fermions
Qi, Hughes, Raghu and Zhang, PRL, 2009
Topological superconductors and superfluids
The BCS-BdG model for 2D equal spin pairing  model of 2D TI by BHZ
where p+=px+ipy. The edge Hamiltonian is given by:
forming a pair of Majorana fermions. Mass term breaks T symmetry=>
topological protection!
Qi, Hughes, Raghu and Zhang, PRL, 2009
Schnyder et al, PRB, 2008
Kitaev
Roy
Tanaka, Nagaosa et al, PRB, 2009
Sato, PRB, 2009
Probing He3B as a topological superfluid (Chung and Zhang,
2009)
The BCS-BdG model for He3B  Model of the 3D TI by Zhang et al
Surface Majorana state:
Qi, Hughes, Raghu and Zhang, PRL, 2009
Schnyder et al, PRB, 2008
Kitaev
Roy
Chiral superconductor obtained from QAH+SC (Qi, Hughes
and SCZ, 2010)
General reading
For a video introduction
of topological insulators
and superconductors,
see
http://www.youtube.
com/watch?v=Qg8Yu
-Ju3Vw
Summary: discovery of new states of matter
Crystal
Quantum Spin Hall
Magnet
s-wave superconductor
Topological insulators