Topological Physics in Band Insulatorsmele/qcmt/ppts/TIpublic.pdf · Topological Physics in ... DRL...
Transcript of Topological Physics in Band Insulatorsmele/qcmt/ppts/TIpublic.pdf · Topological Physics in ... DRL...
Electronic States of MatterElectronic States of MatterBenjamin Franklin (University of Pennsylvania)
That the Electrical Fire freely removes from Place to Place in and thro' the Substance of a Non-Electric, but not so thro' the Substance of Glass. If you offer a Quantity to one End of a long rod of Metal, it receives it, and when it enters, every Particle that was before in the Rod pushes it's Neighbour and so on quite to the farther End where the Overplus is discharg'd… But Glass from the Smalness of it's Pores, or stronger Attraction of what it contains, refuses to admit so free a Motion. A Glass Rod will not conduct a Shock, nor will the thinnest Glass suffer any Particle entringone of it's Surfaces to pass thro' to the other.
Electronic States of MatterElectronic States of Matter
Conductors & Insulators…
This taxonomy is incomplete…
…“Superconductors” (1911)
Electronic States of MatterElectronic States of Matter
Conductors, Insulators & Superconductors
(Onnes 1911)
from Physics Today, September 2010
Electronic States of MatterElectronic States of Matter
Conductors, Insulators & Superconductors
BCS mean field theory ofthe superconducting state (1957)
Univ. of Penn. (1962-1980)
Soliton theory of doped polyacetylene (1978)
Electronic States of MatterElectronic States of Matter
Conductors, Insulators & Superconductors…
This taxonomy is still incomplete…
…“Topological Insulators” (2005)
Band Insulators (orthodoxy)Band Insulators (orthodoxy)
Examples: Si, GaAs, SiO2, etc
Because of the “smallness of its pores”
Band Insulators (atomic limit)Band Insulators (atomic limit)
Examples: atoms, molecular crystals, etc.
“Attraction for what it contains”
Transition from covalent to atomic limitsTransition from covalent to atomic limits
Weaire & Thorpe (1971)
Tetrahedral semiconductors
Modern view: Gapped electronic states Modern view: Gapped electronic states are equivalentare equivalent
Kohn (1964): insulator is exponentially insensitivity to boundary conditions
weak coupling strong coupling “nearsighted”, local
Postmodern: Gapped electronic states are distinguished by topological invariants
Topological StatesTopological StatesTopological classification of valence manifold
Examples in 1D lattices: variants on primer
† †1 1
2 cos
n n n nn
k
H t c c c c
t ka
empty
saturated
?
Cell Doubling (Cell Doubling (PeierlsPeierls, SSH), SSH)
21 2
21 2
0( )
0
ika
ika
t t eH k
t t e
2( ) ( )2
H k H ka
smooth gauge
( ) ( )H k h k 1 2
2
cos 2sin 2
0
x
y
z
h t t kah t kah
Su, Schrieffer, Heeger (1979)
Project onto Bloch SphereProject onto Bloch Sphere
2 21 2 1 2( ) 2 cos2h k t t t t ka
( ) ( )H h k d k
Closed loop Retraced path
2 1t t 1 2t t
Formulation as a BerryFormulation as a Berry’’s Phases Phase
1 11 1;2 2i ie e
Im |
ie
d
: A phase0 : B phase Closed loop
Retraced path
1 †; ( 1)nn n
nC H C H C c c
Domain WallsDomain Walls
Particle-hole symmetry
on odd numbered chain
self conjugate state on one sublattice
Continuum ModelContinuum Model
4 4
4 4
0 0ˆ ˆ
20 0
i i
F x yi i
e eH q iv m
a xe e
1 2m t t “relativistic bands” back-scattered by mass
2 2 2E m v q
Mass inversion at domain wallMass inversion at domain wall
0
0( )( ) exp1
x m xx A dxv
Sublattice-polarized E=0 bound state
vm
Jackiw-Rebbi (1976)
Backflow of filled Fermi seaBackflow of filled Fermi sea
Fractional depletion of filled Fermi sea
Occurs in pairs(topologically connected)
† †1 1
† † †1 1( 1) ( ) ( 1)
n n n nn
n nn n n n n n
n n
H t c c c c
c c c c c c
HeteropolarHeteropolar LatticeLattice
modulate bond strength staggered potential
Rice and GM (1982)
High symmetry casesHigh symmetry cases0, 0
0, 0
0, 0
Continuously tunes the domain wall charge
* ( , )vQ ne Q
One Parameter CycleOne Parameter Cyclee.g. modulate the staggered on-site potential
Charge is shifted periodicallyin a reversible cycle
Two Parameter CycleTwo Parameter Cycleionicity and chirality are nonreciprocal potentials
Nonreciprocal cycle enclosing a point of degeneracy withChern number: 1 ( , )
4 k tS
n dk dt d k t d d
ThoulessThouless Charge PumpCharge Pump
( , ) ( , )H k t T H k t
,
2
2
k kt T t
k tS
eP A dk A dk
e F dk dt ne
with no gap closure on path
n=0 if potentials commuteThouless (1983)
Quantized Hall ConductanceQuantized Hall Conductance
2
2 2 xy
xy
xy
R dt j R dt E Q
h Q nee
enh
Adiabatic flux insertion through 2DEG on a cylinder:
TKNN invariantis the Chern number
Laughlin (1981), Thouless et al. (1983)
Electronic States of MatterElectronic States of Matter
Topological Insulators
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 …insensitive to disorder because their directionality is correlated with spin.
2005 Charlie Kane and GMUniversity of Pennsylvania
Electron spin admits a topologicallydistinct insulating state
Electronic States of MatterElectronic States of Matter
This state is realized in three dimensional materials where spin orbit coupling produces a bandgap “inversion.”
It has boundary modes (surface states) with a 2D Dirac singularity protected by time reversal symmetry.
Bi2Se3 is a prototype.
.Hasan/Cava (2009)
Topological Insulators
The dispersion of a free particle in 2D..
…is replacedby an unconventional E(k) relation on thegraphene lattice
……. it has a critical electronic state. it has a critical electronic state
The low energy theory is described byThe low energy theory is described byan effective mass theory for an effective mass theory for masslessmassless electronselectrons
( ) • ( )Bloch Wavefunction Wavefunction s at K r
eff FH r iv r ( ) ( )
NOTE: Here the “spin” degree of freedom describes the sublatticepolarization of the state, called pseudospin. In addition electrons carrya physical spin ½ and an isospin ½ describing the valley degeneracy.
It is a massless Dirac Theory in 2+1 Dimensions
D.P. DiVincenzo and GM (1984)
A continuum of structures all with √3 x √3 period hybridizes the two valleys
Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry
Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry
Kekule0
'0
ix
ix
eH
e
BN
0'
0z
z
H
Gapping the Dirac PointGapping the Dirac PointCharge transfer from broken
inversion symmetry
Gapping the Dirac PointGapping the Dirac PointOrbital currents from modulated flux
(Broken T-symmetry)
Gauged second neighbor hopping breaks T. “Chern insulator” with Hall conductance e2/h
FDM Haldane “Quantum Hall Effect without Landau Levels” (1988)
FDMH
0'
0z
z
H
FDMH
0'
0z
z
H
1 2
21 2
1 ( , ) 14 k k
S
n d k d k k d d
Topological ClassificationTopological Classification
2
xyeh
“Chern Insulator” with (has equal contributionsfrom two valleys)
2 ' 0 ''
( ) cos sin
2 cos cos sin sin
n x n y zn
n n zn
H k t k a k a M
t k b k b
Orthodoxy: Spectrum Gapped only for Broken Symmetry States
Crucially, this ignores the electron spin
:na triad of nearest neighbor bond vectors
' :nb triad of directed “left turn”
second neighbor bond vectors
Breaks P
Breaks TBreaks e-h symmetry
Coupling orbital motion to the electron spin
SOH s V p
( ) ( )V r V r T Microscopic
Lattice model † † ( )SO m n n m m nH i r r
Spin orbit field Bond vector
Intersite hopping with spin precession
0, 0xy z ˆ ˆ( )SO z z effH s n p p s n p a
0effa d
/ 2
0effa d
† †2
† †
2 † †
cos
sin
i in m m n
n m m n
n m m n
t e c c e c c
c c c ct
i c c c c
Preserve mirror symmetry with a parallel spin orbit field
Generates a spin-dependent Haldane-type mass (two copies)
SO SO z z zs
Coupling orbital motion to the electron spin
Mass Terms (amended)Mass Terms (amended)
z
z z
z z zs x z y y xs s
,x x x y Kekule: valley mixing
Heteropolar (breaks P)
Modulated flux (breaks T)
Spin orbit (Rashba, broken z→-z)
Spin orbit (parallel)**This term respects all symmetries and is therefore present, though possibly weak
spinless
For carbon definitely weak, but still important
Topologically different statesTopologically different states
1 2
21 2
1 ( , )4 k k
S
n d k d k k d d
Charge transfer insulator Spin orbit coupled insulator
0n 1 ( 1) 0n
Topology of Chern insulatorin a T-invariant state
Boundary ModesBoundary Modes
Ballistic propagation through one-way edge state
Counter propagating spinpolarized edge statesIntrinsic SO-Graphene
model on a ribbon
Quantum Spin Hall EffectQuantum Spin Hall EffectIts boundary modes are spin filtered
propagating surface states (edge states)
Symmetry ClassificationSymmetry Classification
Conductors: unbroken state1
Insulators: broken translational symmetry:bandgap from Bragg reflection2
Superconductor: broken gauge symmetry
Topological Insulator ?
1possibly with mass anisotropy
2band insulators
Prospects
Late 1950’s: The band theory of solids introduces novel quantum kinematics (“light” electrons, “holes” etc.) Blount, Luttinger, Kane, Dresselhaus, Bassani…
Mid 1980’s: Identification of pseudo-relativisticphysics at low energy (graphene and its variants)DiVincenzo, Mele, Semenoff, Fradkin, Haldane…
Post 2000: Topological insulators and topological classification of gapped electronic states. Kane, Mele, Zhang, Moore, Balents, Fu, Roy, Teo, Ryu, Kitaev, Ludwig…
Looking forward: Topological band theory as a new materials design principle.(developing)
Study “all things practical and ornamental”