POEM: Physics of Emergent Materials

Transcription

1 POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Reference: Bernevig Topological Insulators and Topological Superconductors Tutorials: May 24, 25 (2017)

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3 Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction atomic SOC band SOC: dresselhaus and rashba symmetries: time reversal, inversion, mirror 2.Berry Phase and Topological Invariant two level system graphene + different mass terms + spin 3.The many Hall effects integer qhe and chern #

4 DAY1 Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction atomic SOC band SOC: dresselhaus and rashba symmetries: time reversal, inversion, mirror 2.Berry Phase and Topological Invariant two level system graphene + different mass terms + spin DAY2 3.The many Hall effects integer qhe and chern #

5 My philisophy. choose simplicity and insight over completeness Toy Problems: Day 1: 1) 1d SOC: spin-momentum locking 2) Two level system (Spin ½ in a magnetic field) and Berry Phase 3) Graphene: dirac points protected by inversion and TR Day 2: 4) 1d SSH [polyacetylene] model and topological invariant 5) Graphene continued: Break inversion: sublattice potential Break TR: Haldane mass

6 with a little help from my friends

7 Basics Basics/Repository Pauli Matrices Symmetries: Inversion, Mirror, Time Reversal Polar vs Axial (pseudo) vector

8 Basics

9 σ τ χ Spin Sublattice Valley

10 Basics Symmetries: Time Reversal Inversion Mirror

11 Definition: Time Reversal (without spin) T : t t TR [H, T] =0 Transformation of various operators under T T ˆxT 1 =ˆx T ˆpT 1 = ˆp TiT 1 = i T ˆLT 1 = ˆL T [ˆx, ˆp]T 1 = Ti T 1 = [ˆx, ˆp] = i Tc j T 1 = c j Tc k T 1 = c k Th(k)T 1 = h( k) K:Complex conjugate T = K T 2 =1

12 Definition: T : t t TR [H, T] =0 T = e iπs y K Time Reversal (with half integer spin) T 2 = 1 T 2 = iσ y iσ y KK = σ 2 y = 1 T Ψ and Ψ are both eigenvectors with eigenvalue E S =( /2) σ e iπσ y/2 = iσ y = ( ) Kramer s degeneracy: every energy level is at least doubly degenerate Tc j T 1 = c j Tc j T 1 = c j Same for dagger operators Th(k)T 1 = h( k)

13 Basics Inversion ~M H ~ = ~ axial x axial = axial

14 Mirror: h(xy) Basics v(xz) Y-mirror x!x y! z!z y L x! L y!l y L z! L x L z v(yz) X-mirror x! y!y z!z x L x!l x L y! L z! L y L z

15 Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction atomic SOC band SOC: dresselhaus and rashba symmetries: time reversal, inversion, mirror 2.Berry Phase and Topological Invariant two level system graphene 3.Hall effects integer qhe and chern #

16 Atom SOC

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18 Atom SOC When is Spin Orbit coupling large? What materials?

19 Atom SOC Transition Metals valence electrons in the d shell

20 s shell L = 0 Transition Metals d shell L = 2 p shell L = 1 Atom SOC f shell L = 3

21 Atom SOC Spherical Symmetry Wavefunction for atomic orbitals For the d shell: radial part angular part (spherical Harmonics) with

22 Atom SOC d Orbitals Atomic orbitals are linear combinations of spherical harmonics

23 Atom SOC Crystal Field Splitting Spherical Symmetry reduced to Octahedral Symmetry Energy degenerate d orbitals degenerate e g orbitals degenerate t 2g orbitals

24 ** Omiting the ħ factor from L. Atom SOC L = 2 Angular Momentum Eigenbasis of L z

25 ** Omiting the ħ factor from L. Atom SOC L = 2 Angular Momentum Orbital basis

26 ** Omiting the ħ factor from L. Atom SOC t 2g Angular Momentum When the t 2g and e g energy levels are sufficiently split and the filling is d 6 or less, we may only consider operators acting in the t 2g subspace and can project all operators into this subspace. Effective L = 1 angular momentum The components satisfy the negative of the usual commutation relations.

27 Atom SOC Spin- Orbit Coupling For t 2g orbitals, the effective L = 1 and S = 1/2 combine to total j = 1/2 and j = 3/2. e g d j = 1/2 L S t 2g crystal field splitting j = 3/2 L S

28 Atom SOC Spin- Orbit Assisted Mott Insulators U l Similar Co, Rh, Ir: 5 valence electrons in the d shell Sr 2 XO 4 tetragonal BCC crystal structure Yet. Sr 2 CoO 4 (Mott Insulator) Sr 2 RhO 4 (metal) Sr 2 IrO 4 (insulator)

29 E DOS E Mott Insulator: Sr 2 CoO 4 Atom SOC t 2g j = 1/2 j = 3/2 L S L S SOC assisted Mott Insulator: Sr 2 IrO 4

30 Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction atomic SOC band SOC: dresselhaus and rashba symmetries: time reversal, inversion, mirror 2.Berry Phase and Topological Invariant two level system graphene 3.Hall effects integer qhe and chern #

31 My philisophy. choose simplicity and insight over completeness Toy Problems: Day 1: 1) 1d SOC: spin-momentum locking 2) Two level system (Spin ½ in a magnetic field) and Berry Phase 3) Graphene: dirac points protected by inversion and TR Day 2: 4) 1d SSH [polyacetylene] model and topological invariant 5) Graphene continued: Break inversion: sublattice potential Break TR: Haldane mass

32 Bands SOC Spin-orbit coupling in bands Example in 1d: spin-momentum locking

33 Bands SOC Spin momentum coupling Each eigenstate is doubly degenerate: Kramer s degeneracy

34 Degeneracy between up and down spins is lifted by SOC Except at specific TRIM: time reversal invariant momenta e.g. p=0 Bands SOC

35 Bands SOC

36 h(k) = cos(k) 0 sin(k) z B x = cos(k) 0 + ~ d(k).~ d x (k) = B d z (k) = Bands SOC sin(k) B =0, =0 B =0, =1 (d x (k),d z (k)) Eigenvector corresponding to lower energy + Eigenvector corresponding to upper energy

37 B =0.1, =1 Bands SOC + No Winding: hence topologically trivial

38 Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction atomic SOC band SOC: dresselhaus and rashba symmetries: time reversal, inversion, mirror 2.Berry Phase and Topological Invariant two level system graphene + mass terms 3.Hall effects integer qhe and chern #

39 Graphene topological quantum matter quantum criticality at Dirac point Graphene

40 Graphene

41 Bands at the chemical potential are derived from p z orbitals Graphene

42 Graphene v F c/350 ~v F = 3 2 ta

43 Graphene Material à Honeycomb Graphene

44 Graphene

45 Graphene

46 Graphene

47 K K

48 Tight-binding model with nearest neighbor hopping on a honeycomb lattice Graphene i A, j B are nearest-neighbor sites, respectively in sublattice A and B (colored red and blue) and σ is a spin index that we suppress Here α is a sublattice index

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50 My philisophy. choose simplicity and insight over completeness Toy Problems: Day 1: 1) 1d SOC: spin-momentum locking 2) Two level system (Spin ½ in a magnetic field) and Berry Phase 3) Graphene: dirac points protected by inversion and TR Day 2: 4) 1d SSH [polyacetylene] model and topological invariant 5) Graphene continued: Break inversion: sublattice potential Break TR: Haldane mass

51 B σ z ± = = ± ± σ x ± = = σ y ± = = ±i ( ) 1 θ =0 χ + = 0 θ = π/2,φ=0 χ + = 1 2 ( 1 1 ) χ = ( 0 1 ) χ = 1 2 ( 1 1 )

52 Apply to graphene: Hamiltonian at momentum k can be written in terms of Pauli matrices in sublattice space A =" B =# τ i for i=1,2,3 : Pauli matrices in sublattice basis d Berry Formally, d(k) in sublattice basis is analogous to magnetic field acting on spin

53 Berry d 1 =+ ~ d 2 = ~ d

54 My philisophy. choose simplicity and insight over completeness Toy Problems: Day 1: 1) 1d SOC: spin-momentum locking 2) Two level system (Spin ½ in a magnetic field) and Berry Phase 3) Graphene: dirac points protected by inversion and TR Day 2: 4) 1d SSH [polyacetylene] model and topological invariant 5) Graphene continued: Break inversion: sublattice potential Break TR: Haldane mass

55 Dirac points protected by TR and inversion symmetry

56 Proof: TR and I protect Dirac points Graphene

57 The Full tight binding Hamiltonian can be expanded around each valley Dirac nodes at Keep only low energy states Expand H k near K + and K -

58 Two independent valleys with Dirac cones Each block can be written in terms of Pauli matrices in sublattice basis h K+ +q = 3 2 ta(q xτ x + q y τ y )= d + ( q). τ h K +q = 3 2 ta( q xτ x + q y τ y )= d ( q). τ Include valley index H q = v F (q x χ z τ x q y τ y ) for i=1,2,3 : Pauli matrices in sublattice basis χ i for i=1,2,3 : Pauli matrices in valley basis

59 Graphene = 0 q x iq y q x + iq y 0 = 0 q x iq y q x + iq y 0

60 Check

61 Graphene

62 Graphene

63 Dirac points are protected when both TR and Inv are present

64 Graphene ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ 0 h / 0 0 h / h / 0 0 h / 0 c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ Graphene where h q = q 8 + i q <

65 Break inversion Sublattice potential Dirac points protected by inversion and time reversal symmetry Break TR complex nn hopping (Color: Berry curvature = Berry flux density)

66 Introduce Dirac mass by breaking symmetry Graphene To introduce mass to the Dirac cones, need to break either inversion or time-reversal symmetry Potential energy difference between sublattices breaks inversion (and C 6 rotation, etc.). This introduces uniform d z (k).

67 Graphene ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ 0 h / 0 0 h / h / 0 0 h / 0 c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ Graphene where h q = q 8 + i q < With sublattice potential ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ m *+ h / 0 0 h / m * m *+ h / 0 0 h / m *+ c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,

68 Introduce Dirac mass by breaking symmetry Graphene To introduce mass to the Dirac cones, need to break either inversion or time-reversal symmetry Next nearest neighbor hopping with imaginary amplitude breaks time-reversal symmetry. This introduces valley dependent d z (k).

69 Graphene ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ 0 h / 0 0 h / h / 0 0 h / 0 c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ Graphene where h q = q 8 + i q < ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ m *+ h / 0 0 h / m * m *+ h / 0 0 h / m *+ c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ With sublattice potential ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ m > h / 0 0 h / m > m > h / 0 0 h / m > c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ With Haldane mass where m > = λ >

70 Graphene With Haldane mass ' H = # c K& 'q,* / ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ m > h / 0 0 h / m > m > h / 0 0 h / m > c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q, Kane- Mele: With Haldane mass σ D where m > = λ > ' H = # c K& 'q,* ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ ' c K& 'q,* ' c K& 'q,+ ' c K, 'q,* ' c K, 'q,+ / m > h / h / m > m > h / h / m > m > h / h / m > m > h / h / m > c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ c K& 'q,* c K& 'q,+ c K, 'q,* c K, 'q,+ spin spin

71 σ τ χ Spin Sublattice Valley

72 .sheer poetry