Report

Journal Club of Topological Materials (2014) If you raised your hand you’re in the wrong place!! Show of hands, who here is familiar with the concept of topological insulators? 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 Physics 2, no. 1 (2011): 31-53. 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 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 Current = 1 μA Magnetic Field = 18 T Temperature = 1.5 K • Fractional Quantum Hall Effect (FQHE) discovered in 1982 • Phase transitions do not involve symmetry breaking • 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” 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 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 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 Odd under time reversal Constant • Non-zero spin Hall conductance in the presence of time-reversal symmetry Even under time reversal Even under time reversal Constant • 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 Phenomenology of the Quantum Spin Hall Effect • QSHE as a “topologically” distinct phase • “Fractionalization” at the boundary Spinless Spinless 1D1D chain chain 2 =21= +11+1 Spinful Spinful 1D1D chain chain 4 =42= +22+2 Impurity Impurity QHQH QSQ HSH • “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) The QSHE in HgTe Quantum Wells • Review of basic solid state physics • What does spin-orbit coupling do? • What does time-reversal symmetry imply? Kramers pair states • Kramers pairs well defined even when spin is not conserved • What does inversion symmetry imply? • What do both time-reversal and inversion symmetries imply? The QSHE in HgTe Quantum Wells • Banstructure of bulk CdTe • s-like (conduction) band Γ6 and p-like (valence) bands Γ7 and Γ8 with (right) and without (left) turning on spin-orbit interaction 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 • With spin-orbit interaction Γ8 splits into the Light Hole (LH) and Heavy Hole (HH) bands away from the Γ point • The split-off band Γ7 shifts downward The QSHE in HgTe Quantum Wells • Banstructure of bulk HgTe • s-like (conduction) band Γ6 and p-like (valence) bands Γ7 and Γ8 with (right) and without (left) turning on spin-orbit interaction 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 -4 -4 • The Γ8 splits into LH and HH like CdTe except the LH band is inverted • The ordering of LH band in Γ8 and Γ6 bands are switched The QSHE in HgTe Quantum Wells • Quantum Well (QW) fabrication • Molecular Beam Epitaxy (MBE) grown HgTe/CdTe quantum well structure • Confinement in (say) the z-direction • Transport in the x-y plane Band gap of barrier z 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 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 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 BHZ Model 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 The QSHE in HgTe Quantum Wells • The BHZ Model • Solutions • Normalization condition 0.04 0.03 0.02 • Bulk dispersion 0.01 0 -0.01 • Surface dispersion -0.02 -0.03 where s labels Kramers pairs -0.04 The QSHE in HgTe Quantum Wells • The BHZ Model • Using Landauer-Büttiker formalism for an nterminal device • For the helical edge channels we expect • For a 2-point transport measurement between terminals 1 and 4 • A -flux tube threaded into a QS spin-charge separation. (Qi & Zh The QSHE in HgTe Quantum Wells proposal, Ran, Vishwanath, Lee) • The BHZ Model • Start from decoupled case 2 H=+e /h • If the transport is dissipationless where is the 2 resistance coming from? H=-e /h • In QSHE don’t we have spin currents of e2/h + e2/h = 2e2/h and charge currents of e2/h – e2/h = • Flux threading in quantum Hall 0? 1981) • 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 Annu. Rev. Condens. Matter Phys. 2011.2:31-53. Downloaded f by California Institute of Technology on 12/13/13. For bulk HgTe and its consequences for the HgTe/CdTe QW subband structure. Therefore, one should first verify whether band inversion in the HgTe/CdTe system exists. A striking manifestation of this is a so-called re-entrant QH effect (26) that has been experimentally observed (Section 1.3) (see Figure 5). The peculiar band structure ofinverted HgTe/CdTe QWs gives rise The QSHE in HgTe Quantum Wells • The BHZ Model • For normal ordering of bands the Landau levels will get further apart as B increases • For inverted bandstructures Landau levels will cross at a certain B a dQW = 40 Å b dQW = 150 Å 0 5 100 40 50 20 E meV–1 E meV–1 0 0 –50 –20 –100 0 5 10 15 B T–1 10 15 B T–1 5 • Only inverted bandstruc- Figure Bulk Landau levels ( fan diagram ) for an HgTe/CdTe quantum well (QW) in a perpendicular magnetic field B. Trivial insulator ( d < d ): No level crossing occurs as a function of B, and for a fixed Fermi energy E in the tures will reenter the quan- (a) B ¼ 0 gap, the Hall conductance s is always zero. ( b) Quantum spin Hall insulator with d > d : There is a crossing at some critical field B ¼ B , and for a fixed E in the B ¼ 0 gap, a conduction or valence band tum Hall states when B field level Landau level eventually crosses E , giving rise to a re-entrant quantum Hall effect with s e /h. increases c F xy c c F 2 F xy www.annualre views.org The Quantu m Spin Hal l Effect Theory of the Helical Edge State Spinless Spinless 1D1D chain chain • 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 backscattering paths • Clockwise and anticlockwise rotation of spin pick up ±π phase leading to destructive interference 2 =21= +11+1 Spinful Spinful 1D1D chain chain 4 =42= +22+2 Impurity Impurity QHQH QSQ HSH Theory of the Helical Edge State • The physical description of edge state protection works only for single pair of edge states • With (say) two forwardmovers and two backward-movers backscattering is possible without spin flip • Robust or non-dissipative edge transport requires odd number of edge states Stability of the Helical Liquid: Disorder and Interactions • Only two TR invariant non-chiral interactions can be added forward scattering term Two-particle backscattering or “Umklapp” 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) 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: Bosonize Nx 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 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 inhomogeneities can trap bulk electrons which may then interact with the edge electrons K<1 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 Stability of the Helical Liquid: Disorder and Interactions • Static magnetic impurity breaks local TRS and opens a gap • Quantum impurity Kondo effect 1. At high temperature (T) conductance (G) is log 2. For weak Coulomb interaction (K > 1/4) conductance back to 2e2/h. At intermediate T the G ~ T2(4K-1) due to Umklapp term 3. For strong Coulomb 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 Fractional-Charge Effect and Spin-Charge Separation • Quantized charge at the edge of domain wall o Jackiw-Rebbi (1976) o 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 = m cos(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] Fractional-Charge Effect and Spin-Charge Separation • Two magnetic islands trap the electrons between them like a 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 spin-charge separation. (Qi & Zh proposal, Ran, Vishwanath, Lee) Fractional-Charge Effect and inSpin-Charge •QSH Startinsulators from decoupled case II. Spin-charge separation =+e /h Separation • A -flux tube threaded into a QSH insulator induces =-e /h H 2 2 H • Simplified analysis:spin-charge separation. (Qi & Zhang, see similar (t proposal, Ran, Vishwanath, Lee) o Assume Sz is preserved • Start from decoupled case o QSHE as two copies of QHE 2 • Flux threading in quantum Hall H=+e /h E 1981) • Thread a π (units of ℏ = c = e = 1) flux ϕ j 2 • TRS preserved at ϕ = 0 and π; also, π = –πH=-e /h • Four possible paths for ϕ↑ and ϕ↓: • Current density from E||: • Flux threading in quantum Hall system. (Laughlin PRB • Net charge flow: 1981) 3D Topological Insulators • Introduction • 2D topological insulator 1D edge states • Dirac-like edge state dispersion (a) (a) •Vacuum What happens in(c) 3D? up spin down spin • Surface dispersion is a Dirac cone, like graphene (c) down spin up spin Dirac point Γ Bulk Valence Band k=0 kx k Helical spin polarization 2D Dirac cone ky Surface Brillouin zone (b) Energy Bulk Conduction Band E (b) Energy (d) up spin down spin down spin 1D? Nothing! (c Vacuum spin • 3D up topological insulator 2D surface states • What happens in 2D Topological Insulator (c Vacuum 2D Topological Insulator 2D Topological Insulator (d Bulk Conduction Band down spin (d Bulk up spinBand Conduction Dirac point down spin up spin Dirac point Bulk Valence Band k=0 k k=0 k Bulk Valence Band 3D Topological Insulators • Topological band theory • Difficult to evaluate ℤ2 invariants for a generic band structure • Consider the matrix • At the TRIM B(Γi) is antisymmetric; we can define • Topological invariant • Trivial: (–1)ν = +1 and Non-trivial: (–1)ν = –1 2D 2D • “Dimensional increase” to 3D • Weak TI: (–1)ν = +1 and Strong TI: (–1)ν = –1 3D 3D 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 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