Download - Topological Physics in Band Insulators
![Page 1: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/1.jpg)
Topological Physics inTopological Physics inBand InsulatorsBand Insulators
Gene MeleDepartment of Physics
University of Pennsylvania
![Page 2: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/2.jpg)
A Brief History of A Brief History of Topological InsulatorsTopological Insulators
What they are
How they were discovered
Why they are important
![Page 3: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/3.jpg)
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.
![Page 4: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/4.jpg)
Electronic States of MatterElectronic States of Matter
Conductors & Insulators…
This taxonomy is incomplete…
…“Superconductors” (1911)
![Page 5: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/5.jpg)
Electronic States of MatterElectronic States of Matter
Conductors, Insulators & Superconductors
(Onnes 1911)
from Physics Today, September 2010
![Page 6: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/6.jpg)
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)
![Page 7: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/7.jpg)
Electronic States of MatterElectronic States of Matter
Conductors, Insulators & Superconductors
W.P. Su, J.R. Schrieffer & A.J. Heeger (Penn);GM & M.J. Rice (Penn/Xerox)
Topological Defects in (CH)x
![Page 8: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/8.jpg)
Electronic States of MatterElectronic States of Matter
Conductors, Insulators & Superconductors…
This taxonomy is still incomplete…
…“Topological Insulators” (2005)
![Page 9: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/9.jpg)
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
![Page 10: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/10.jpg)
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 the prototype.
.Hasan/Cava (2009)
Topological Insulators
![Page 11: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/11.jpg)
Modern view: Gapped (TModern view: Gapped (T--invariant)invariant)electronic states are equivalentelectronic states are equivalent
Kohn (1964): insulator is exponentially insensitive to boundary conditions
weak coupling strong coupling “nearsighted”, local
Postmodern: Gapped electronic states are distinguished by topological invariants
![Page 12: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/12.jpg)
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)
![Page 13: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/13.jpg)
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
![Page 14: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/14.jpg)
1 †; ( 1)nn n
nC H C H C c c
Domain WallsDomain Walls
Particle-hole symmetry
on an odd N chain
self conjugate state on one sublattice
![Page 15: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/15.jpg)
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
![Page 16: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/16.jpg)
Key Ideas from SSH ModelKey Ideas from SSH Model
Topological classification of insulating state(winding number)
Change of topology at interface(the mass changes sign!)
Universal spectral signature from topology(self conjugate, spin-charge reversed etc.)
Kohn: “Insulating states are equivalent”
Postmodern: “Actually, interfaces between inequivalent insulating states are equivalent”
![Page 17: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/17.jpg)
GrapheneGraphene: the Parent Phase: the Parent Phase
![Page 18: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/18.jpg)
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
![Page 19: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/19.jpg)
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)
![Page 20: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/20.jpg)
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
![Page 21: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/21.jpg)
Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry
Kekule0
'0
ix
ix
eH
e
![Page 22: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/22.jpg)
BN
0'
0z
z
H
Gapping the Dirac PointGapping the Dirac PointCharge transfer from broken
inversion symmetry
![Page 23: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/23.jpg)
Gapping the Dirac PointGapping the Dirac PointOrbital currents from a modulated flux
(requires breaking 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
![Page 24: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/24.jpg)
Topological ClassificationTopological Classification
1 2
21 2
1 ( , ) 04 k k
S
n d k d k k d d
![Page 25: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/25.jpg)
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)
![Page 26: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/26.jpg)
Mass Terms For Single Layer Mass Terms For Single Layer GrapheneGraphene
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 itis therefore present, though possibly weak
spinless
For carbon it is definitely weak, but still important
with spin
![Page 27: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/27.jpg)
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 (which restores T)
SO SO z z zs
Coupling orbital motion to the electron spin
![Page 28: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/28.jpg)
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 insulator occursin a time reversal invariant state
![Page 29: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/29.jpg)
Topologically Protected Topologically Protected Boundary ModesBoundary Modes
Ballistic propagation through one-way edge state
Counter propagating spinpolarized edge statesIntrinsic SO-Graphene
model on a ribbon
![Page 30: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/30.jpg)
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
![Page 31: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/31.jpg)
Symmetry ClassificationSymmetry ClassificationOrdinary insulators and topological insulators are distinguishedby a two-valued (even-odd) surface index.
Kramers Theorem: T-symmetry requires E(k,) =E(-k,)
But at special points k and -k are identified (TRIM)
even: ordinary (trivial) odd: topological
Kane and GM (2005)
![Page 32: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/32.jpg)
With inversion symmetryWith inversion symmetryOrdinary insulators and topological insulators are distinguishedby a two-valued ( = 0,1) bulk index.
1
( 1)N
aa
a mm
(parity eigenvalues, 1)
Fu, Kane and GM (2007)
![Page 33: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/33.jpg)
Symmetry ClassificationSymmetry ClassificationOrdinary insulators and topological insulators are distinguishedby their strong coupling limits.
“nearsighted” topological“nearsighted”, local
![Page 34: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/34.jpg)
2D Surface of a (strong) 3D TI2D Surface of a (strong) 3D TI
Discrete Kramers points: Each surface has a single Dirac node, with a chiral partner on the opposite face.
Gapped by an exchange field(breaks T) or by a pairfield (via proximity effect)
![Page 35: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/35.jpg)
Single Layer versus Single Layer versus BilayerBilayer GrapheneGraphene
Single layer Bilayer Asymmetric Bilayer
Bias reversal producesa mass inversion
![Page 36: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/36.jpg)
Single Layer versus Single Layer versus BilayerBilayer GrapheneGraphene
Single layer Bilayer Asymmetric Bilayer
& valley polarized modes:Martin et al, (2008);
F. Zhang and GM (2012)
![Page 37: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/37.jpg)
The interlayer coherence scale forThe interlayer coherence scale forthe Bernal stacked the Bernal stacked bilayerbilayer
Single layer Bernal Bilayer
collapsescollapses ininfaulted faulted multilayersmultilayers (!)(!)
e.g. M. Sprinkle et al.PRL 103, 226803 (2009) (ARPES)
![Page 38: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/38.jpg)
The interlayer coherence scale for The interlayer coherence scale for the Bernal stacked the Bernal stacked bilayerbilayer
Single layer Bernal Bilayer
![Page 39: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/39.jpg)
LayerLayer--polarized Dirac modespolarized Dirac modes
1 11 1
w
0 11 0abc
1 00 1aac
isotropic(canonical) anisotropic (-)
anisotropic (+)
This is a symmetry-protected band crossing.
It is controlled by the residual lattice registry in the moire-averaged interlayer coupling in its continuum(small angle) limit.
![Page 40: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/40.jpg)
SymmetrySymmetry--protected band crossingprotected band crossing
anisotropic (+)
anisotropic (-)
new DP
![Page 41: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/41.jpg)
Band insulators: a timelineBand insulators: a timeline
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, Ludwig, Schnyder..
Looking forward: Topological band theory as a new materials design principle.(developing)
Study “all things practical and ornamental”
![Page 42: Topological Physics in Band Insulators](https://reader031.vdocuments.us/reader031/viewer/2022012103/616a0b3711a7b741a34e30b4/html5/thumbnails/42.jpg)
Collaborators:Collaborators:Charlie Kane
David DiVincenzoLiang FuPetr Kral
Andrew RappeMichael RiceSaad ZaheerFan Zhang