Journal Club of Topological Materials (2014)

1. Journal Club of Topological Materials (2014)

2. If you raised your hand you’re in the wrong place!! Show of hands, who here is familiar with the concept of topological insulators?

3. The Quantum Spin Hall Effect Tejas Deshpande Joseph Maciejko, Taylor L. Hughes, and Shou-Cheng Zhang. “The Quantum Spin Hall Effect.” Annual Reviews of Condensed Matter Physics2, no. 1 (2011): 31-53.

4. Introduction • Ginzburg-Landau Theory of Phase Transitions • Classify phases based on which symmetries they break • Rigorous definition of “symmetry breaking”: ground state does not possess symmetries of the Hamiltonian • Example: classical Heisenberg model • Ordered phase characterized by local order parameter • Phases Defined by Symmetry Breaking • Rotational and Translational: Crystalline Solids (continuous to discrete) • Spin Rotation Symmetry: Ferromagnets and Antiferromagnets • U(1) gauge symmetry: Superconductors

5. Introduction • “Topological” Phases • Integer Quantum Hall Effect (IQHE) discovered in 1980 • Topological or “global” order parameter  Hall conductance quantized in integral units of e2/h • Fractional Quantum Hall Effect (FQHE) discovered in 1982 • Phase transitions do not involve symmetry breaking Current = 1 μA Magnetic Field = 18 T Temperature = 1.5 K • Experimental implications of “topological order” • Number of edge states equal to topological order parameter (Chern number) • Edge states robust to all perturbations due to “topological protection”

6. Introduction • Topological Protection • Current carried only by chiral edge states • Chiral edge states robust to impurities • No tunneling between opposite edges • FQHE • FQHE with (1/m)e2/h (m odd) Hall conductance gives rise to bosonic quasiparticles • Example: FQHE with m = 3 has quasiparticles with 3 flux quanta attached • Chern-Simons theory is the low energy effective field theory

7. Introduction • Road to Topological Insulators (TIs) • IQHE without a magnetic field: Haldane model • Observation of the “spin Hall effect” Spin conductance Occupations of Light-Hole (LH) and Heavy-Hole (HH) bands

8. Phenomenology of the Quantum Spin Hall Effect • Classical spin vs. charge Hall effect • Charge Hall effect disappears in the presence of time-reversal symmetry Even under time reversal Even under time reversal Odd under time reversal Even under time reversal Constant Constant • Non-zero spin Hall conductance in the presence of time-reversal symmetry • Does the quantum version of the spin Hall effect exist? • Yes! Kane and Mele proposed the quantum spin Hall effect (QSHE) in graphene and postulated the Z2 classification of band insulators

9. Phenomenology of the Quantum Spin Hall Effect • QSHE as a “topologically” distinct phase • “Fractionalization” at the boundary • “Topological” in the sense that the electron degrees of freedom are spatially separated • Mechanism of spatial separation: • QHE  External magnetic field (time-reversal breaking) • QSHE  intrinsic spin-orbit coupling (time-reversal symmetric)

10. The QSHE in HgTe Quantum Wells • Review of basic solid state physics • What does spin-orbit coupling do? Kramers pair states • What does time-reversal symmetry imply? • Kramers pairs well defined even when spin is not conserved • What does inversion symmetry imply? • What do both time-reversal and inversion symmetries imply?

11. The QSHE in HgTe Quantum Wells • Banstructure of bulk CdTe • s-like (conduction) band Γ6and p-like (valence) bands Γ7and Γ8 with (right) and without (left) turning on spin-orbit interaction • With spin-orbit interaction Γ8splits into the Light Hole (LH) and Heavy Hole (HH) bands away from the Γ point • The split-off band Γ7shifts downward

12. The QSHE in HgTe Quantum Wells • Banstructure of bulk HgTe • s-like (conduction) band Γ6and p-like (valence) bands Γ7and Γ8 with (right) and without (left) turning on spin-orbit interaction • The Γ8splits into LH and HH like CdTe except the LH band is inverted • The ordering of LH band in Γ8and Γ6bands are switched

13. The QSHE in HgTe Quantum Wells • Quantum Well (QW) fabrication • Molecular Beam Epitaxy (MBE) grown HgTe/CdTequantum well structure • Confinement in (say) the z-direction • Transport in the x-y plane z Band gap of barrier EF E Band gap of QW • L = 600 μm and W = 200 μm • Gate voltage (VG) used to tune the Fermi level (EF) in HgTe quantum well

14. The QSHE in HgTe Quantum Wells • Topological phase transition • QW sub-bands invert for well thickness d > 6.3 nm • Intersection of the first electron sub-band with hole sub-bands

15. The QSHE in HgTe Quantum Wells • The Bernevig-Hughes-Zhang Model • Hamiltonian with QW symmetries • Components • Elegant Hamiltonian form • Break translational symmetry in the y-direction

16. BHZ Model

17. The QSHE in HgTe Quantum Wells • The BHZ Model • Numerical diagonalization? • Try ansatz • Writing • Plugging in explicit expressions and multiplying by Γ5 we get • Since

18. The QSHE in HgTe Quantum Wells • The BHZ Model • Solutions • Normalization condition • Bulk dispersion • Surface dispersion • where s labels Kramers pairs

19. The QSHE in HgTe Quantum Wells • The BHZ Model • Using Landauer-Büttikerformalism for an n-terminal device • For the helical edge channels we expect • For a 2-point transport measurement between terminals 1 and 4

20. The QSHE in HgTe Quantum Wells • The BHZ Model • If the transport is dissipationless where is the resistance coming from? • In QSHE don’t we have spin currents of e2/h + e2/h = 2e2/h and charge currents of e2/h – e2/h = 0? • Answer 1: dissipation comes from the contacts. Note that transport is dissipationless only inside the HgTe QW • Answer 2: We do measure charge conductance! The existence of helical edge channels is inferred from charge transport measurements

21. The QSHE in HgTe Quantum Wells • The BHZ Model • For normal ordering of bands the Landau levels will get further apart as Bincreases • For inverted bandstructures Landau levels will cross at a certain B • Only inverted bandstruc-tureswill reenter the quan-tum Hall states when B field increases

22. Theory of the Helical Edge State • The concept of “helical” edge state  states with opposite spin counter-propagate at a given edge • QH protected by “chiral” edge states; QSH edge states protected due to destructive interference between all possible back-scattering paths • Clockwise and anticlockwise rot-ation of spin pick up ±π phase leading to destructive inter-ference

23. Theory of the Helical Edge State • The physical description of edge state protection works only for single pair of edge states • With (say) two forward-movers and two back-ward-movers backscattering is possible without spin flip • Robust or non-dissipative edge transport requires odd number of edge states

24. Stability of the Helical Liquid: Disorder and Interactions • Only two TR invariant non-chiral interactions can be added Two-particle backscattering or “Umklapp” term forward scattering term • We can “bosonize” the Hamiltonian • Boson to fermion field operators • The forward scattering term simply renormalizes the parameters K and vF • Combined with Umklapp term we get (opens a gap at kF = π/2)

25. Stability of the Helical Liquid: Disorder and Interactions • Total Hamiltonian Umklapp term • RG analysis  Umklapp term relevant for K < 1/2 with a gap: • Interactions can spontaneously break time-reversal symmetry • TR odd single-particle backscattering: • BosonizeNx and Ny . For gu < 0 fixed points at • For gu < 0, Ny is the (Ising-like) ordered quantity at T = 0 • Due to thermal fluctuations TRS is restored for T > 0 • For mass order parameter Ny is disordered + TR is preserved with a gap

26. Stability of the Helical Liquid: Disorder and Interactions • Total Hamiltonian Umklapp term • Two-particle backscattering due to quenched disorder Gaussian random variables • The “replica trick” in disordered systems shows disorder relevant for K < 3/8 • Nx and Ny show glassy behavior at T = 0 with TRS breaking; TRS again restored at T > 0 • Where would all these interactions come from? locally doped regions? Band bending? • But edge states are immune to electrostatic potential scattering • Potential inhomogeneitiescan trap bulk electrons which may then interact with the edge electrons K < 1

27. Stability of the Helical Liquid: Disorder and Interactions • Static magnetic impurity breaks local TRS and opens a gap • Quantum impurity  Kondo effect: • Doing the “standard” RG procedure we get flow equations

28. Stability of the Helical Liquid: Disorder and Interactions • Static magnetic impurity breaks local TRS and opens a gap • Quantum impurity  Kondo effect • At high temperature (T) conductance (G) is log • For weak Coulomb interaction (K > 1/4) conductance back to 2e2/h. At intermediate T the G ~ T2(4K-1) due to Umklapp term • For strongCoulomb interaction (K < 1/4) G = 0 at T = 0 due to Umklapp. At intermediate T the G ~ T2(1/4K–1) due to tunneling of e/2 charge

29. Fractional-Charge Effect and Spin-Charge Separation • Quantized charge at the edge of domain wall • Jackiw-Rebbi (1976) • Su-Schrieffer-Heeger (1979) • Helical liquid has half DOF as normal liquid  e/2 charge at domain walls • Mass term ∝ Pauli matrices  external TRS breaking field • Mass term to leading order • Current due to the mass field • For m1 = mcos(θ), m2= m sin(θ), and m3 = 0 • Topological response  net charge Q in a region [x1,x2] at time t = difference in θ(x,t) at the boundaries • Charge pumped in the time interval [t1, t2]

30. Fractional-Charge Effect and Spin-Charge Separation • Two magnetic islands trap the electrons between them likea quantum wire between potential barriers • Conductance oscillations can be observed as in usual Coulomb blockade measurements • Background charge in the confined region Q (total charge) = Qc (nuclei, etc.) + Qe (lowest subband) • Flip relative magnetization  pump e/2 charge • Continuous shift of peaks with θ(B) • AC magnetic field drives current

31. Fractional-Charge Effect and Spin-Charge Separation • Simplified analysis: • Assume Sz is preserved • QSHE as two copies of QHE • Thread a π (units of ℏ = c = e = 1) flux ϕ • TRS preserved at ϕ= 0 and π; also, π = –π • Four possible paths for ϕ↑ and ϕ↓: • Current density from E||: • Net charge flow:

32. 3D Topological Insulators • Introduction • 2D topological insulator  1D edge states • Dirac-like edge state dispersion • What happens in 3D? • 3D topological insulator  2D surface states • Surface dispersion is a Dirac cone, like graphene • What happens in 1D? Nothing!

33. 3D Topological Insulators • Topological band theory • Difficult to evaluate ℤ2invariants for a generic band structure • Consider the matrix • At the TRIM B(Γi) is antisymmetric; we can define • Topological invariant • Trivial: (–1)ν2D = +1 and Non-trivial: (–1)ν2D = –1 • “Dimensional increase” to 3D • Weak TI: (–1)ν3D = +1 and Strong TI: (–1)ν3D = –1

34. 3D Topological Insulators • Simplified topological invariant expression • With inversion symmetry rewrite δi as where ξ2m(Γi) = ±1 is the parity eigenvalue of the 2mth band at Γi) and ξ2m= ξ2m–1 are Kramers pairs • Recall BHZ model • Gap closing (phase transition) • k = (0, 0)  M = 0 • k = (π, 0) and (0, π)  M = 4B • k = (π, π)  M = 8B

35. Conclusion and Outlook • The quantum spin Hall effect (QSHE) • Phenomenology • Design of quantum wells in the QSHE regime • Explicit solution of Bernevig-Hughes-Zhang (BHZ) model • Experimental verification using transport • Properties of the “2D topological insulator” • Theory of helical edge states • Effects of interactions and disorder • Fractionalization and spin-charge separation • Introduction to 3D topological insulators • Topological Band Theory (TBT) • Topological Invariant of the QSHE