topological physics in band insulators iiimele/qcmt/ppts/windsorlec...summary of third lecture: this...
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Gene MeleUniversity of Pennsylvania
Low energy models for real materials
Topological Physics inTopological Physics inBand Insulators IIIBand Insulators III
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Summary of Second Lecture: A two-valued integer index distinguishesconventional and topological insulators. A change of this index at a boundarybetween insulators signals the existence of symmetry-protected Dirac modespropagating along the interface.
Summary of Third Lecture: This physics is realized in a family of 2D and 3D crystalline solids. Goal: connect band theoretic analysis of the bulk and low energy representation of its protected interface states.
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Gapping the Gapping the GrapheneGraphene Dirac PointDirac Point
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
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0 0z s z z zs
Counterpropagating spin-filterededge states
Edge: Hamiltonian undergoes asingle mass inversion (at K or K’) in each spin sector
topological conventional
SpinSpin--filtered edge statesfiltered edge states
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CommentsComments
The energy scale for this effect is very small (carbon is a light atom)
But the physics is robust (topological)
Hybrid structures:“enhance” spin orbit field from adsorbed heavy metallic species
Spin-orbit coupled semiconductors:Band inversion occurs in narrow gapsemiconductors with strong s-o coupling
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B. A. Volkov and O. A. Pankratov, JETP Lett. 42, 178 (1985)
History: special interface states at a History: special interface states at a bandband--inverting junctioninverting junction
Pb1-xSnxTe alloys: interface states controlled by asymptotics
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Band inversion in IIBand inversion in II--VI semiconductorsVI semiconductors
“conventional”“inverted”
J=3/2bonding p
J=1/2antibonding s
J=3/2bonding p
J=1/2antibonding s
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Recent history: level ordering in a Recent history: level ordering in a IIII--VI (001) quantum wellVI (001) quantum well
k=0 states are indexed by axial symmetry
1/ 2 6 8, 1/ 2 , 1/ 2
3/ 2 8 , 3 / 2
With low energy space spanned by 4x4
*
( , ) 0( , )
0 ( , )x y
x yx y
h k kH k k
h k k
“Decoupled 2x2 blocks”
Konig et al (Molenkamp group) (2007)
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Representation as two state systemRepresentation as two state system
, 2 2( ) ( )I ( )x yh k k k d k
2
20
( )( ) ( , , )x y
k C Dkd k Ak Ak M Bk
average
mixing gap
Mass inversion occurs for a “wide” quantum well
Bernevig, Hughes and Zhang (BHZ, 2006)
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with anomalous Landau quantizationwith anomalous Landau quantization
conventional inverted
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2D QSHE is observed via ballistic 2D QSHE is observed via ballistic transport through its edge modestransport through its edge modes
Molenkamp group (2007, 2008)
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2D experimental status2D experimental status
The realization of 2D QSHE in HgTe quantum wellsrequires strong spin-orbit coupling and broken cubicsymmetry in a thin heterostructure.(challenging fabrication, T~30 mK, B~ 10 T)
The “decorated graphene” strategy remains an unsolved experimental challenge.
Amazingly, this physics occursspontaneously in 3D materials that are readilysynthesized and measurable at RT.
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Examples in 3DExamples in 3DAlloys of BixSb1-x : Band inversion at three L points. Z2=-1 for range of x
Bi2Se3 and related tri-chalcogenides. Stacked quintuple layers.
Mass reversal → interface state → spin texture
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A kindergarten metaphorA kindergarten metaphor1D Edge of a 2D Topological Insulator is like this
P.G. Silvestrov, P.W. Brouwer and E.G. Mischenko “On the structureof surface states of topological insulators” arXiv:1111.3650v3A. Medhi and V.B. Shenoy “Continuum Theory of Edge States of TopologicalInsulators” arXiv:1202.3863v1F. Zhang, C.L. Kane and GM “Surface States of Topological Insulators”arXiv:1203.6382v1
2D Boundary of a 3D Topological Insulator is more like
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2D topological insulators admits asimple theory of the 1D helical edge states
Outline of calculation in 2D :
( ) ( 0)
1. linearize 4 band model:
y y yy
HH k H k kk
1 2
( 0)
( ) ( )
( ) ( )
2. with node at x=0 admits Kramers degenerate
evanescent solutions:
with
y
x x
y
H k
x a e e x
x x
3. Bulk model prescribes y yy
Hvk
the protected edge state is robustbut this result is fragile
S.C. Zhang group (2009)
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Level ordering in BiLevel ordering in Bi22SeSe33
atoms bonds parity crystal field spin orbit
Layered structure 3R m
bandinversion
H. Zhang et al (Shoucheng Zhang group, 2009)
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BiBi22SeSe33 as TI Prototypeas TI PrototypeEight bulk time reversal invariant momenta: , Z, F(3), L(3).
Band inversion is confined to small momenta near (occupiedbands “buried” at seven other TRIM).
Hasan/Cava (2009)
ARPES: Single symmetry-protected Dirac cone measured on the (001) face
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Graphene Topological Insulator
Pairs of DP’s(chiral partners)
Single DP(partner on opposite face)
Gapped by T-preservingz potential (breaks P)
Ungapped by any T-preserving(protected Kramers pair)
Spin and valley degeneracieshide odd half-integral QHE
Odd half-integral QHEon a single face (TI=1/4 graphene)
Weak trigonal warping Strong hexagonal warping
Small quantum corrections in MR Weak antilocalization in bothsurface and bulk channels
g ~ 2 g ~ 30
N/A Face-dependent spectra:topological continuity via side faces
(helical)h v p ˆ (twisted, chiral)h v n p
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Topological Insulator Surface StatesTopological Insulator Surface StatesBulk Boundary Correspondence:The properties of the edge modes are determinedby the bulk symmetries of the materials joined at the interface. (i.e. mass reversal)Single Valley Physics:Bi2Se3 –type materials have a single band inversion near . The long wavelength theory is a gradientexpansion within a single valley (minimal model).
Bulk Anisotropy: The bulk Hamiltonian is not isotropic(Bi2Se3 is a layered material). Surface propertiesare therefore strongly face-dependent.
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Bulk HamiltonianBulk Hamiltonian
0 1 2( , )zH k k H H H
2 3( ), ( ), ,Symmetries:
z yC x C z P T iK
+ -z z z
Four band model (Bi(p ), Se(p ), J = 1/2):
0 0 0 2 0sgn(- ) Band Inversion, zH c m m
1 z z y y x x y xH v k v k k Mixing ( k)
2 2 2 22 Band Curvaturez z z z zH c k c k m k m k
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More preciselyMore precisely
This 3D extension of this idea is subtle.
Most properties of the edge modes (energy in gap, spin structure, influence of bulk anisotropy, sensitivity to surface localized potentials, etc.) are accessible in the fourband theory but require a specification of the boundary condition that terminates the bulk Hamiltonian.
For an ideal termination of a 3D TI this is given by a “topological boundary condition.” For nonideal terminations this is augmented by a two-parameter family of surfacelocalized potentials constrained by P and T symmetries.
Fan Zhang, Kane and GM (2012)
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Topological Boundary Condition ITopological Boundary Condition I
Topological interface is characterizedby a mass inversion at the TI surface. Four-band degrees of freedom boostedto mass scale M.
For M>>m0 exterior wf specifiestermination condition for bulkevanescent waves.
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Surface State HamiltonianSurface State Hamiltonian surf y x x yH v k k The primitive theory of the (001) surface:
Inherits important quadratic terms that break e-h symmetry andshift the Kramers point from the midgap
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Crystal Face DependenceCrystal Face DependenceThese effects are both strong and crystal face dependent because ofthe anisotropy of the bulk Hamiltonian
Vacuum (isotropic)
TI (R3m)
top face
side face
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001 Surface (cleavage plane)
1 2General surface has structure! S S
In representation midgap solution issymmetric under -rotations
1
2
2 2
2 2
, ,
, ,
cos( cos ) ( sin )
sin
( cos ) ( sin )
( -like)
( -like)
x y y y y x z
x z z y z x z
z
z
z
S
S
vv v
v
v v
Structure of four band modelStructure of four band modelFan Zhang, Kane and GM (2012)
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FaceFace--dependent Dirac physicsdependent Dirac physics
And the energy of the DP
Anisotropy in bulk SO couplingdetermines spin texture of its protected surface mode
self doping at step edges
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Hexagonal warping and spin textureHexagonal warping and spin texture 3 3 zsurf y x x yH v k k k k
Symmetry allowed couplingwarps FS (seen) and tips
Spin out of plane on cleavage surface
Chen (2009), Fu (2010), Gedik group (2011)
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Topological Boundary Condition IITopological Boundary Condition II
Nonideal interface projected intofour-band representation has an additional surface potential thatrotates the wf of the target state.
J-R boundary condition &mismatch condition fromsurface potential specifiestermination of its bulk evanescent states.
Fan Zhang, Kane and GM (2012)
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Quantum well states from band bending
Chemical shift:differential shift of DP
within gap
Field effect:band bending and
quantum confinement
Chen et al (2012),Bahramy et al. (2012)
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For the surface of a strong topological insulatorthe SHAPE COUNTS
All STI faces support a Dirac nodebut the orbital and spin texture are
nonuniversal and crystal face-dependent
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Kramers TRIM Degeneracy
Symmetry Breaking by FM exchange coupling
Is removed by z
But is shifted by
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1D Chiral Mode along domain wall
Reversing exchange field = mass inversion
1 2Algebra for any noncleavage plane: S S
2 2 1
2
2 2 1
( ) ( )
( ) ( )
x z zx x x x
yy y y
z x zz z z z
S S SS
S S S
Mass inversion occurs at “edges” for uniform exchange field
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Edge induced 1D Chiral Modes
Mass inversion at side “edges”
(001) exchange field
Bi2Se3/Fe4Se7intergrowth
Cava (2012)
Fan Zhang, Kane and GM (2012)
sphere
slab
layered
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Some References:
Review Article: M.Z. Hasan and C.L. Kane Rev. Mod. Phys. 82, 3045 (2010)
BHZ Model: B.A. Bernevig, T. Hughes and S.C. ZhangScience 314, 1757 (2006)
Four Band Model for Bi2Se3: H. Zhang et al, Nature Physics 5, 438 (2009)
Four Band Models (second generation)Fan Zhang, C.L. Kane and E.J. Mele arXiv:1203.6382