Topological Physics in Band Insulators. Gene Mele DRL 2N17a

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1 Topological Physics in Band Insulators Gene Mele DRL 2N17a

2 Electronic States of Matter Benjamin 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 entring one of it's Surfaces to pass thro' to the other.

3 Electronic States of Matter Conductors & Insulators This taxonomy is incomplete Superconductors (1911)

4 Electronic States of Matter Conductors, Insulators & Superconductors (Onnes 1911) from Physics Today, September 2010

5 Electronic States of Matter Conductors, Insulators & Superconductors BCS mean field theory of the superconducting state (1957) Soliton theory of doped polyacetylene (1978) Univ. of Penn. ( )

6 Electronic States of Matter Conductors, Insulators & Superconductors This taxonomy is still incomplete Topological Insulators (2005)

7 Band Insulators (orthodoxy) Examples: Si, GaAs, SiO 2, etc Because of the smallness of its pores

8 Band Insulators (atomic limit) Examples: atoms, molecular crystals, etc. Attraction for what it contains

9 Transition from covalent to atomic limits Tetrahedral semiconductors Weaire & Thorpe (1971)

10 Modern view: Gapped electronic states are 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

11 Topological States Topological classification of valence manifold Examples in 1D lattices: variants on primer H t c c c c k n n1 n n n1 2t cos ka saturated empty?

12 Cell Doubling (Peierls( Peierls,, SSH) H ( k) smooth gauge t 0 t e 1 2 2ika t t e ika 2 H ( k ) H ( k) 2a hx t1 t2 cos 2ka H ( k) h( k) h t sin 2ka Su, Schrieffer, Heeger (1979) h y z 2 0

13 Project onto Bloch Sphere H h( k) d ( k) h( k) t t 2t t cos2ka t 2 t1 1 2 t t Closed loop Retraced path

14 Formulation as a Berry s s Phase ; i i 2 e 2 e Im i e d : A phase 0 : B phase Closed loop Retraced path

15 Domain Walls Particle-hole symmetry 1 C H C H C c c on odd numbered chain ; ( 1) n n n n self conjugate state on one sublattice

16 Continuum Model i i 4 4 e 0 e 0 H q iv ˆ m ˆ i i 2a x e 0 e F x y E m v q relativistic bands back-scattered by mass m t t 1 2

17 Mass inversion at domain wall Sublattice-polarized E=0 bound state x m( x) 0 ( x) Aexp dx v 1 0 v m Jackiw-Rebbi (1976)

18 Backflow of filled Fermi sea Fractional depletion of filled Fermi sea Occurs in pairs (topologically connected)

19 Heteropolar Lattice H t c c c c n n1 n n n1 n ( 1) ( c c c c ) ( 1) c c n n n1 n n n1 n n n modulate bond strength staggered potential Rice and GM (1982)

20 On Bloch Sphere H ( k) h( k) h t ( t ) cos 2ka x h ( t ) sin 2ka h y z 0 0 0

21 High symmetry cases 0, 0 0, 0 0, 0 Continuously tunes the domain wall charge * Q ne Qv (, )

22 One Parameter Cycle e.g. modulate the staggered on-site potential Charge is shifted periodically in a reversible cycle

23 Two Parameter Cycle ionicity and chirality are nonreciprocal potentials Nonreciprocal cycle enclosing a point of degeneracy with Chern number: 1 n dk dt d ( k, t) kd td 4 S

24 Thouless Charge Pump H ( k, t T ) H ( k, t) with no gap closure on path e P Ak dk Ak dk 2 tt t e Fk, t dk dt ne 2 S Thouless (1983) n=0 if potentials commute

25 Quantized Hall Conductance Adiabatic flux insertion through 2DEG on a cylinder: 2 R dt j 2 R dt xye Q xy h e xy Q 2 e n h ne Laughlin (1981), Thouless et al. (1983) TKNN invariant is the Chern number

26 Electronic 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 Charlie Kane and GM University of Pennsylvania Electron spin admits a topologically distinct insulating state

27 Electronic States of Matter Topological Insulators 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. Bi 2 Se 3 is a prototype.. Hasan/Cava (2009)

28 Graphene: : the Parent Phase

29 .. it has a critical electronic state The dispersion of a free particle in 2D.. is replaced by an unconventional E(k) relation on the graphene lattice

30 The low energy theory is described by an effective mass theory for massless electrons Bloch Wavefunction Wavefunction( s) at K ( r ) H ( r ) iv ( r ) eff It is a massless Dirac Theory in 2+1 Dimensions NOTE: Here the spin degree of freedom describes the sublattice polarization of the state, called pseudospin. In addition electrons carry a physical spin ½ and an isospin ½ describing the valley degeneracy. F D.P. DiVincenzo and GM (1984)

31 Gapping the Dirac Point Gapping the Dirac Point Valley mixing from broken translational symmetry A continuum of structures all with 3 x 3 period hybridizes the two valleys

32 Gapping the Dirac Point Valley mixing from broken translational symmetry H ' 0 i x Kekule i e x 0 e

33 Gapping the Dirac Point Charge transfer from broken inversion symmetry H ' 0 z BN 0 z

34 Gapping the Dirac Point Orbital currents from modulated flux (Broken T-symmetry) H ' 0 z FDMH 0 z Gauged second neighbor hopping breaks T. Chern insulator with Hall conductance e 2 /h FDM Haldane Quantum Hall Effect without Landau Levels (1988)

35 Topological Classification 1 (, ) 0 2 n d k d k1 k2 k d 1 k d 2 4 S

36 Topological Classification H ' 0 z FDMH 0 z 1 (, ) 1 e Chern Insulator with xy h 2 n d k d k1 k2 k d 1 k d 2 4 S 2 (has equal contributions from two valleys)

37 a n: b n' : Orthodoxy: Spectrum Gapped only for Broken Symmetry States H ( k) t cos k a sin k a M n n x n y z 2t cos cos k b sin sin k b 2 n' 0 n' z n' Breaks e-h symmetry triad of nearest neighbor bond vectors triad of directed left turn second neighbor bond vectors Breaks T Breaks P Crucially, this ignores the electron spin

38 Microscopic Coupling orbital motion to the electron spin H s V p SO V ( r ) V ( r T ) H i ( r r ) Lattice model SO m n n m m n Spin orbit field Bond vector Intersite hopping with spin precession

39 xy Coupling orbital motion to the electron spin Preserve mirror symmetry with a parallel spin orbit field 0, 0 H s nˆ p p ( s nˆ ) p a z SO z z eff / 2 a a eff eff d 0 d 0 i i t2 e cncm e cmcn t 2 cos cncm cmcn isin c c c c Generates a spin-dependent Haldane-type mass (two copies) SO SO z zsz n m m n

40 Mass Terms (amended) x x, x y Kekule: valley mixing z Heteropolar (breaks P) spinless z z s x z y y x z zsz s 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 For carbon definitely weak, but still important

41 Topologically different states Charge transfer insulator Spin orbit coupled insulator 1 n 0 n 1 ( 1) 0 2 n d k d ( k1, k2) k d 1 k d 2 4 S Topology of Chern insulator in a T-invariant state

42 Boundary Modes Ballistic propagation through one-way edge state Counter Intrinsic propagating SO-Graphene spin polarized model on edge a ribbon states

43 Quantum Spin Hall Effect Its boundary modes are spin filtered propagating surface states (edge states)

44 Symmetry Classification Conductors: unbroken state 1 Insulators: broken translational symmetry: bandgap from Bragg reflection 2 Superconductor: broken gauge symmetry Topological Insulator? 1 possibly with mass anisotropy 2 band insulators

45 Prospects Study all things practical and ornamental 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-relativistic physics 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)