TOPOLOGICAL INSULATORS
(Mattia Gaboardi)
Phase Transitions
Landau Theory (spontaneous broken-symmetry):
●
Crystals: translational and rotational symmetry breaking
● FM/AFM: rotational symmetry braking of spin space
● Liquid Crystal: rotational but not translational
● Superconductors: gauge symmetry breaking
●
Phase Transitions
●
1980 (QHE): possibility to have transitions that do not
become involved symmetry breaking (the behaviour
does not depend on geometry)
●
Topological states of matter:
BULK insulators, conductor outside:
–
–
Edge (2D)
Surface (3D)
●
Spin-up/spin-down separation
●
Importance of band structure (topology)
●
Protected states on EDGE/surface
cups, donuts and knots
g=0
g=1
g=3
TOPOLOGY:
● Study of figure's properties which
don't changes when we do a
deformation without:
● splitting
● overlapping
● gluing
Topological Invariant = quantity that does not change under continuous
deformation
Band Insulator
The insulating state is the most common state of matter:
● Energy gap between C.B. and V.B.
● Energy gap of one atom is bigger than that of a semiconductor
● Electronic surfaces like topological figures in the Fourier's space
● All conventional insulators are topologically equivalents
● Therefore: Insulator is “equal” to void (energy-gap due to pairs production of electronspositrons)
QUESTION: all the electronic states with a gap are equivalent to the void?
NO!
Integer Quantum Hall Effect (IQHE)
It is the simplest system topologically ordered:
● Electrons confined in 2D interface between two
semiconductors in strong magnetic field
● Lorentz force: k independent Landau levels
Low T
● High magnetic field
● Pure sample
●
E m =ℏ  c m1/2
von Klitzing et al., 1980
IQHE
 xy =
h
ne 2
IQHE
e2
 xy=n
h
n=0
n=
1
2
∇
×
Ak
,
k
d
k
∫
x
y
2  B.Z.
A=⟨ uk∣−i ∇ k∣u k ⟩
n≠0
surface
edge
Landau levels: band insulator
● Hall conductivity (xy)
● Chiral current on edge! No backscatterig!
● n: interpreted as Chern number
(topological invariant)
● topological vision of Hall effect
●
The state responsible for the
QHE does not break any
symmetry, but defines a
topological phase: some of
the fundamental properties of
the system are insensitive to
smooth variations of the
parameters of the material.
IQHE
●
n is called TKKN invariant (Thouless, Kohmoto, Nightingale, Nijis; 1982) and for
IQHE, n=1
●
●
The topological index distinguishes a simple insulator (n=0) from a QH state
(n≠0).
The quantum of σxy is a topological quantum number: it depend only by
electronic structure of bulk, not by surface.
–
“Holographic image” of the bulk
●
Chern's number (topological invariant): Berry's phase
●
TKKN demonstrates that σxy has the same shape of n
–
n cannot change if the hamiltonian change smoothly
EDGE states of an insulator cannot be destroyed by defects or impurities because they
depend solely on the topological state of the bulk (I cannot destroy them without first
destroying the topological state of the Hilbert's space of bulk).
● Applications in quantum computers (protection from dephasing) and spintronics
IQHE
●
●
Interesting but...
●
High B
●
Low T (cryogenics)
●
“perfect” crystals
Breaking of time-reversal symmetry
B
Hall conductivity is odd
under time inversion
GRAPHENE
●
●
Simple example of QHE in band theory (graphene in
periodic field)
Haldane (1988): fictitious magnetic field:
–
<B(r)> = 0
–
B(r) with same symmetry of the lattice
●
B(r)=0 :
zero gap (2 Dirac points)
●
B(r)≠0 :
energy-gap
–
●
Gapped Dirac particles
Not a normal insulator. Prototype of 2D-QSH system
K'=-K
GRAPHENE
●
●
Degeneracy at Dirac points protected by:
●
Parity (spatial invariance), P
●
Time reversal symmetry, T
I can remove degeneracy by breaking one of this 2 symmetries
●
P: 2 different atoms for cell
●
T: by applying magnetic field (Haldane)
–
–
–
–
B zero on average, with full symmetry of the lattice
Energy-gap
This state is not associated with
an insulator: is a QH system with
2
e
n = 1:
 xy=
h
For a T-invariant system Dirac points must come in pairs fermion doubling theorem
SPIN-ORBIT Interaction
●
Relativistic effect
●
Magnetism in matter (magnetic anisotropy)
●
Internal effective magnetic field (Haldane, 1988)
●
●
Seen as combination of 2 opposite fields playing on 2 different spin
states
Counterpropagating spin-polarized current
Topological Insulator
●
Hall conductivity is ODD under time inversion
●
●
Topologically nontrivial states occur only when T is broken
Kane, Mele (2005): Spin Orbit interaction allows a different topological class
of insulating band structures when T symmetry is unbroken!
●
●
●
●
●
T-symmetry is represented by antiunitary operator, Θ (Θ2 = -1)
Kramers' theorem: “all eigenstates of a T-invariant hamiltonian are at
least twofold degenerate”
A T-invariant Bloch hamiltonian must satisfy:
−1
 H  k  =H −k 
If there are bound states near the edge: the Kramers' theorem requires
they are twofold degenerate at the T-invariant momenta kx=0 and π/2.
Away the edge: S.O. Interaction will remove this degeneracy:
TRIVIAL METAL: the
surface states cross
the Fermi level an
even number of times
TOPOLOGICAL
INSULATORS: the
surface states cross
the Fermi level an
odd number of times
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
●
●
●
This state was originally theorized to exist in graphene and 2D
semiconductors system with a uniform strain gradient (Kane & Mele, 2005)
Predicted (Bernevig, Hughes, and Zhang, 2006) and observed (König et al.,
2007) in HgCdTe quantum well structures
Degeneracy at the Dirac point in graphene is protected by inversion and Tsimmetry. But we ignored the spin of electrons!
●
●
Hamiltonian decouples into 2 independent hamiltonians for the UP and
DOWN spins
The resulting theory is simply two copies of the Haldane's model with
opposite signs of the Hall conductivity for UP and DOWN spins
●
T-reversal flips both the spins and σxy.
●
In an applied Field:
–
Hall conductivity is thus ZERO, but there is a quantized Spin-Hall
conductivity σsxy= 2e2/h
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
z
x
y
-B
e2
 xy=n =0
h
B
1D spin-liquid
This electrons form an unique 1D conductor that is essentially half
of a ordinary 1D Fermi liquid
n=0 !
Is the only
topological
invariant (TKKN
invariant)
QSH edge states are “spinfiltered”: UP spins
propagate in one direction;
DOWN spins propagate in
the other. “Helical states”, in
analogy with helicity of a
particle.
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
●
●
Ordinary conductors (UP and DOWN electrons propagate in both directions)
are fragile due to Anderson's localization
QSH edge states cannot be localized even for strong disordered!
●
●
●
●
●
It follows that unless T-symmetry is broken, an incident electron is
transmitted perfectly across the defect (at T=0K: ballistic transport)
For T>0K inelastic backscattering processes are allowed, which will lead
to a finite conductivity
Graphene is made out of carbon (weak S.O. Interaction)
●
●
Scattering involves flipping the spin
Energy gap in graphene will be very small ( 10-3meV )
I have to search heavier elements!
(Bernevig, Hughes, and Zhang, 2006): quantum well of Hg1-xCdxTe (family of
semiconductor with strong S.O. interaction)
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
●
●
CdTe : normal ZnS semiconductor
–
Valence states: p-like symmetry
–
Conduction states: s-like symmetry
HgTe :
–
p levels rise above the s levels, leading to an
inverted band structure
HgTe of d thickness between CdTe layers:
●
●
●
d<6.3nm : 2D electronic states bound to the quantum well
have the normal band order
d>6.3nm : the 2D bands invert. Quantum phase transition
between the trivial insulator and the quantum spin Hall
insulator.
This can be understood simply in the approximation that
the system has inversion symmetry. In this case, since
the s and p states have opposite parity the bands will
cross each other at dc without an avoided crossing. Thus,
the energy gap at d=dc vanishes
heavy
light
split-off
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
Narrow quantum well
(d<6.3nm): insulator
L=20μm
Inversion
regime
L=1μm
Existence of edge states
of the QSHI
● Sample II: finite
temperature scattering
effects
● Sample III and IV exhibit
conductance 2e2/h
associated with the top and
bottom edges
●
(gate voltage)
d>6.3nm
(inverted regime)
Tunes the Fermi
level through the
bulk energy gap
Quantum Spin-Hall Effect (QSHE)
(2D topological insulator or QSH-Insulator, QSHI)
d<dc
d=dc
d>dc
3D Topological Insulators
Fu-Kane (2007):
●
●
●
New type of systems which don't exhibit QSHE (theory)
Chern numbers (νi) like “order parameters” (from their knowledge i go back
to phase)
4 different topological invariants (instead of one): 16 different type of
insulators
●
●
If ν0=ν1=ν2=ν3=0 : 2D Topological insulator (QSHI)
Surface conducting states (instead of edge)
bulk
more...
If I look the
spins, i see
that they
rotate around
the Fermi
surface!
3D Topological Insulators
●
●
●
The surface states of a 3D-T.Ins. can be labeled with a 2D crystal
momentum (kx,ky).
There are 4 T-invariant points (Γ1,2,3,4) in the surface B.Z., where surface
states must be Kramers degenerate.
●
Away from this points the S.O. Interaction will lift the degeneracy
●
Kramers points form 2D Dirac points in the surface band structure
The simplest 3D-T.Ins. may be constructed by staking layers of 2D-QSHI
●
●
●
This is called “WEAK” T.Ins., and a possible Fermi surface is:
This state has ν0=0 and (ν1,ν2,ν3)=(h,k,l), describing the orientation of the
layers
Unlike 2D-QSHI, T-symmetry does not protect these surface states
3D Topological Insulators
●
ν0=1 identifies a distinct phase, called a “STRONG” T.Ins.
●
●
●
●
It cannot be interpreted as a descendent of the 2D-QSHI
Infact, ν0 determines whether an EVEN or ODD numbers of Kramers
points is enclosed by the surface Fermi circle
In a STRONG T.Ins.: surface Fermi circle encloses an ODD number of
Kramers degenerate Dirac points!
Similar to graphene, but:
●
Graphene: 4 Dirac points
●
STRONG T.Ins.: single Dirac point !?
–
This appears to violate the fermion doubling theorem...
–
Partner Dirac points reside on opposite surfaces!
3D Topological Insulators
●
Surface states of a strong T.Ins. form a unique 2D topological metal
●
●
●
Ordinary metal (2D Fermi gas): up and down spins at every point of
Fermi surface
Strong T.Ins.: the surface states are not spin degenerate
–
T-symmetry requires that states at momenta k and -k have opposite
spin
–
So, the spin must rotate with k around the Fermi surface!
–
When an electron circles a Dirac point, its spin rotates by 2π: πBerry phase
Electrons at the surface cannot be localized even for strong disorder as
long as the bulk energy gap remains intact!
Inversion of
chirality
Kramer Point
The first 3D-T.Ins.: Bi1-xSbx
●
Bi1-xSbx: Semiconducting alloy with interesting thermoelectric properties
Pocket of holes
●
Pure Bi: semimetal with strong S.O. Coupling
●
Pure Sb:
●
●
La,b: band derived from
antisymmetric/symmetric
orbitals
When x=0.04 the gap between La and Ls closes and a massless 3D
Dirac point is realized!
Bi is the trivial (0;000) class while Sb is the (1;111) class.
–
●
Pocket of electrons
Since for x=0.4 Bi1-xSbx is on the Sb side of the band inversion
transition it will be (1;111).
Problem: charge transport experiments (which were successful for
QSHI), are problematic in 3D materials because it is difficult to separate
the surface contribution to the conductivity from that of the bulk
Angle Resolved Photo-emission
Spectroscopy (ARPES)
●
Ideal tool for probing the topological character of the surface states
●
●
It uses a photon to eject a photo-electron from a crystal and then
determines the surface or bulk electronic structure from an analysis of
the momentum of the emitted electron
It can also measure the spin orientation on the Fermi surface!
2D or 3D
excitations
Bi1-xSbx
5 DIRAC CONES!
Map of the energy of the occupied
surface electronic states as a
function of k:
Surface states are nondegenerate
and strongly spin polarized
Surface Fermi surface
(111) surface
projection
D. Hsieh et al. (2008)
Bi1-xSbx
Spin-ARPES
map of the
surface state
measured at
Fermi level has a
spin-texture
B.Z.
●
●
●
●
Fourier
Transform of the
observed pattern
FFT
Direct lattice
Spin polarization rotates by 360° around centre of Fermi surface
Spin texture on Fermi surface provides a first direct evidence for the π-Berry
phase
The topological surface states are expected to be robust in the presence of
nonmagnetic disorder and immune from Anderson localization
This due to the fact that T-symmetry forbids the backscattering between
Kramers pairs at k and -k
Second generation materials:
Bi2Se3, Bi2Te3 and Sb2Te3
●
●
Surface structure of Bi1-xSbx was rather complicated and the band-gap was
small
Searching of larger band-gap and simpler surface spectrum
New materials are not alloys: more
control on purity
● Bi Se :
2
3
● Single Dirac cone
● Larger bulk band-gap
● Change in chirality above Dirac point
● T-symmetry preserved
● Topological behaviour at room
temperature!
● No external magnetic fields needed
● Also impure crystals
●
Second generation materials:
Bi2Se3, Bi2Te3 and Sb2Te3
●
Many of theoretical proposals require the chemical potential to lie at or near
the surface Dirac point!
●
●
●
●
This make the density of carriers highly tunable by applied electric field
and enables application also in microelectronics
Generally is not so (unlike in graphene)!
By appropriate chemical modifications, however, the Fermi level can be
controlled
Hsieh et al, (2009): doping the bulk with a small concentration of Ca; the
surface was doped with NO2 to place Fermi level at Dirac point
Exotic Broken Symmetry Surface
Phase
●
1980: integer plateaus are seen experimentally in IQHE
●
●
1983: Fractional plateaus are seen experimentally (Fractional QHE) with
only odd denominators
●
●
Explanation: nearly free electrons with ordinary fermionic statistics
Explanation: interacting electron liquid that hosts “quasiparticles” with
fractional charge and fractional “anyonic” statistics
1989: a plateau is seen when 5/2 Landau levels are filled
●
Explanation: interacting electron liquid that hosts “quasiparticles” with
non-Abelian statistics (anyons)
(+1): bosons
(-1) : fermions
2D: phase
●
In 3D particles are restricted to be bosons or fermions; in 2D
“quasiparticles” can be observed which obey statistics ranging
continuously between Fermi–Dirac and Bose–Einstein statistics
(anyons)
Exotic Broken Symmetry Surface
Phase
●
Interface between 3D-T.Ins. and 3D-SPC may allow the creation of an
'emergent' “quasiparticle”: Majorana fermion excitation (proposal)
●
●
If a vortex line runs from the SPC into the T.Ins., then a zero-energy
Majorana fermion is trapped in the vicinity of the vortex core.
●
●
●
●
Like any other metal, the T.Ins. become SPC (proximity effect)
It has quantum numbers that differ from those of an ordinary electron
–
Bounded state composed by: 1 Electron + even number of fluxons
–
It is its own antiparticle (a Majorana fermion is essentially half of an
ordinary spinless Dirac fermion). a=a†
–
It is electrically neutral
Also predicted in Sr2RuO4 and 2D structures that combine SPC, FM and
strong S.O. Coupling
Non-Abelian quantum statistic
Majorana fermions are one step towards a topological quantum computer
(exceptionally protected from errors)
Exotic Broken Symmetry Surface
Phase
●
Topologically protected from local sources of decoherence
Conclusions
●
●
●
●
●
T.Ins. are closely related to the Dirac electronic structure of
graphene (relativistic particles)
Only one Dirac point (only on surface/edge) and no spindegeneracy
Electrons are never completely reflected when scattered (not
localized)
Fermi level in T.Isn. does not have any reason to sit at the Dirac
point; however, it can be tuned with chemical modifications
Possibility to generate new particles (Majorana fermions)
●
2 separeted Majoranas = 2 degenerate states (1 qubit)
●
2N separeted Majoranas = N qu-bits
References
Hasane, Kane; Rev. Mod. Phys, vol. 82 (2010)
Kane, Moore; Physics World (2011)
T.,K.,K.,N.,; PRL, vol. 49, 6 (1982)
Xiao Liang Qi, Physics Today, 33-38 (2010)
Kane, Mele; PRL, vol. 95, 226801 (2005)
König et al.; Science, 318, 766 (2007)
B. Andrei Bernevig, et al.; Science, vol. 314, 1757 (2006);
Stern; Nature, vol. 464, 11 (2010)
Haldane; PRL, vol. 61, 18 (1988)
Kane, Mele; Science, vol. 314 (2006)
WIKIPEDIA!
http://www.youtube.com/watch?v=2kk_CcRXEMY
“God made the bulk.
Surfaces were invented
by the devil”
W. Pauli