(Mattia Gaboardi) - unipr.it precedenti... · splitting overlapping ... In an applied Field: – Hall conductivity is thus ZERO, ... Away from this points the S.O. Interaction will

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 InsulatorThe 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 electrons-positrons)

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

Em=ℏc m1/2

● Low T● High magnetic field● Pure sample

von Klitzing et al., 1980

IQHE

xy=h

ne2

IQHE

xy=n e2

hn=

12∫B.Z.

∇×Ak x , k yd2 k

A=⟨uk∣−i ∇ k∣uk ⟩

● Landau levels: band insulator● Hall conductivity (xy)● Chiral current on edge! No back-scatterig!● 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.

surface

edge

n=0

n≠0

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 smoothlyEDGE 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

BHall 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

n = 1:– For a T-invariant system Dirac points must come in pairs -

fermion doubling theorem

xy=e2

h

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:● 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:

H k −1=H −k

TOPOLOGICAL INSULATORS: the surface states cross the Fermi level an odd number of times

TRIVIAL METAL: the surface states cross the Fermi level an even 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 T-simmetry. 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


B

-B xy=n e2

h=0

n=0 !Is the only topological invariant (TKKN invariant)

QSH edge states are “spin-filtered”: UP spins propagate in one direction; DOWN spins propagate in the other. “Helical states”, in analogy with helicity of a particle.This electrons form an unique 1D conductor that is essentially half

of a ordinary 1D Fermi liquid

yx

z

1D spin-liquid



● 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! ● Scattering involves flipping the spin● 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)

● 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-x

CdxTe (family of

semiconductor with strong S.O. interaction)


● 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 structureHgTe 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 d

c without an avoided crossing. Thus,

the energy gap at d=dc vanishes

heavy

light

split-off


(gate voltage)

Tunes the Fermi level through the bulk energy gap

Narrow quantum well(d<6.3nm): insulator

L=20μm

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

Inversion regime

d>6.3nm (inverted regime)


d<dc d=dc d>dc

3D Topological InsulatorsFu-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=ν

1=ν

2=ν

3=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,k

y).

● 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 (ν

1,ν

2,ν

3)=(h,k,l), describing the orientation of the

layers● Unlike 2D-QSHI, T-symmetry does not protect these surface states

3D Topological Insulators● ν

0=1 identifies a distinct phase, called a “STRONG” T.Ins.

● It cannot be interpreted as a descendent of the 2D-QSHI● Infact, ν

0 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: π-

Berry 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-x

Sbx

● Bi1-x

Sbx: Semiconducting alloy with interesting thermoelectric properties

● Pure Bi: semimetal with strong S.O. Coupling● Pure Sb:

● 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.

– Since for x=0.4 Bi1-x

Sbx 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

Pocket of holes

Pocket of electrons

La,b

: band derived from antisymmetric/symmetric orbitals

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-x

Sbx

(111) surface projection

Map of the energy of the occupied surface electronic states as a function of k:

D. Hsieh et al. (2008)

Surface Fermi surfaceSurface states are nondegenerate and strongly spin polarized

5 DIRAC CONES!

Bi1-x

Sbx

Spin-ARPES map of the surface state measured at Fermi level has a spin-texture

Fourier Transform of the observed pattern

● 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

B.Z. Direct lattice

FFT

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● Bi2Se3:

● 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● Explanation: nearly free electrons with ordinary fermionic statistics

● 1983: Fractional plateaus are seen experimentally (Fractional QHE) with only odd denominators

● 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)

● 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)

(+1): bosons(-1) : fermions2D: phase


● Interface between 3D-T.Ins. and 3D-SPC may allow the creation of an 'emergent' “quasiparticle”: Majorana fermion excitation (proposal)

● Like any other metal, the T.Ins. become SPC (proximity effect)● 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.● 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)


● 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 spin-

degeneracy● 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

ReferencesHasane, 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

http://www.youtube.com/watch?v=2kk_CcRXEMY

“God made the bulk. Surfaces were invented by the devil”

W. Pauli

Documents

(Mattia Gaboardi) - unipr.it precedenti... · splitting overlapping ... In an applied Field: – Hall conductivity is thus ZERO, ... Away from this points the S.O. Interaction will