Topological Kondo Insulators!

Transcription

1 Topological Kondo Insulators! Maxim Dzero, University of Maryland Collaborators: Kai Sun, University of Maryland Victor Galitski, University of Maryland Piers Coleman, Rutgers University Main idea Kondo Insulators Topological insulators Are Kondo insulators topological? Yes!!"# #')(" '(*(%(!,&%!"#$%&'() k z !5 67&8 #')(" '(*(%(!,&% '(*(%(!,&%!"#$%&'()!"#$%&'() k z k z -./ :4!5!$# $# "#! !5 jour-ref: Phys. Rev. Lett. 104, (2010)!

2 increasing localization increasing localization Coleman (2002)! A lot of action takes place on the brink of localization! Non-Fermi Liquid phases Unconventional Superconductivity Hidden Order: URu 2 Si 2 Metal-Insulator transitions

3 Kondo Effect Cu 1-x Fe x A. J. Heeger, in Solid State Physics vol 23 (1969) At high temperatures: free local moment S! k B log 2 At low temperatures: moment is quenched single impurity Kondo effect C! k B T T K

4 Effective mass of quasiparticles is heavily renormalized (20-100x band mass) Fermi surface is well described by the band theory At high temperatures: local moment metals Coleman (2002)! At low temperatures: moments quench to form heavy fermions

5 Kondo Insulators: SmB 6 Magnetic susceptibility flattens out below 100 K Band structure calculations: mixed valence behavior

6 Kondo Insulators: Ce 3 Bi 4 Pt 3 Hundley et al. (1990)! Hybridization gap arises due to interaction between 4f and conduction band electrons Gap is suppressed by doping: disordered Kondo Lattice Real part of the optical conductivity: disappearance of the spectral weight below 100 K. Bucher et al. (1994)!

7 Ce ( f 1 ) in tetragonal crystal field environment strong spin-orbit coupling f 1 : S =1/2, L=3,J= L S =5/2 J =5/2 f Γ 7 f Γ 7 ± = cos β 3/2 sin β ±5/2 ± = sin β 3/2 cos β ±5/2 f Γ 6 ± = ±1/2 Actual position of the Kramers doublets is determined experimentally (XAFS)!

8 Kondo insulators: theory Anderson model: Ĥ = k,α ξ k c kα c kα jα [ ] V ψ jα f jα h.c. jα [ ε (0) f n f,jα U ] f 2 n f,jαn f,jα Hybridization ψ jα = 1 V k Φ ασ (ˆk)e ik x j f-electron! form factor! c kσ f 2 f 1 f 0 kσ Φ(ˆk) Γ 6 α Φ (ˆk) kσ Strong spin-orbit coupling is present on the level of interaction between c- and f-electrons

9 Kondo insulators: theory Anderson model: Ĥ = k,α ξ k c kα c kα jα [ ] V ψ jα f jα h.c. jα [ ε (0) f n f,jα U ] f 2 n f,jαn f,jα ψ jα = 1 V k Φ ασ (ˆk)e ik x j f-electron! form factor! c kσ form factors: [Φ Γk ] ασ = m Γα jm jm kσ tight-binding [Φ Γk ] ασ = m [ 3,3] Γα 3m, σ Z Matrix element between! the Bloch and Wannier states! R 0 Y 3 M ( ˆR)e ik R Spin-orbit coupling is present on the level of Interaction between c- and f-electrons

10 Kondo insulators: theory correlation functions G cc (k,iω) = G ff (k,iω) = [ approximations: iω ξ k [ iω ε (0) f iω ε (0) f V (k) 2 Σ f (k,iω) neglect self-energy dispersion: Kondo limit ignore the physics at high Matsubara frequencies hybridization amplitude Σ f (k,iω) V (k) 2 iω ξ k ] 1 f-level renormalization ] due to Hubbard-U 1 interaction ε f = Z k [ε (0) f Σ f (k, 0)] Z k = Ṽ (k) = ZV (k) [ 1 Σ f (k, ω) ω ] 1 ω=0

11 Mean-field theory: effective Hamiltonian! ξk 1 Ṽ Φ H mf (k) =( ) Γk Ṽ Φ Γk ε f 1 Qualitative description renormalized! position of the! f-level! ω k T T K ω k T T K µ ε (0) f µ ε f

12 Hybridization gap: momentum dependence Tr [ˆΦ Γk ˆΦ Γk ] Γ 7 M J = ±3/2 Γ 6 M J = ±1/2 Γ 7 M J = ±5/2 Nodes along k z Fully gapped! Nodes along k z Linear combination of any of two gapless shapes with the fully gaped one yields non-vanishing hybridization gap. TKI with nodes?

13 Quantum Hall Kondo insulators P. Ghaemi & T. Senthil (2007); M. Dzero et al. (2010)! 2D model of the Kondo insulator m k = H = k ( αv ˆ, αv ˆ, 1 ) 2 (ɛ k ε f ) [ 1 2 (ɛ k ε f ) m k τ ] maps torus (BZ) to a sphere current operator j µ = H k µ σ xy = e2 d 2 k m ( x m y m) 4πh m 3 V chiral mode exists even in the absence of external magnetic field 2D Kondo insulator has a quantized Hall conductivity

14 Topological insulators M. Z. Hasan & C. L. Kane,! Topological Insulators, RMP (2010).! surface states as a function of surface crystal momentum Four bulk Z 2 invariants determine whether the surface states are protected (even or odd # of points) Simplest 3D TI: stack of layers {x, y, z} Surface states can be pushed outside the gap Surface state is topologically protected three indices (Miller indices: orientation of the layers) Strong TI: odd number of Dirac points enclosed by the FS weak TI strong TI

15 Topological insulators L. Fu & C. Kane, Topological Insulators with! Inversion Symmetry, PRB 76, (2007).! Response to a fictitious applied magnetic field 2D: Flux plays the role of the edge crystal momentum 3D: two fluxes corresponding to two components of the surface crystal momentum Z 2 invariants are computed from the parity of the occupied bands: change in time reversal polarization due to changes in bulk Hamiltonian

16 Topological insulators: invariants L. Fu & C. Kane, Topological Insulators with! Inversion Symmetry, PRB 76, (2007).! topological structure is determined by parity at high symmetry points ( ) 1 0 parity P = time-reversal T = 0 1 ( ) iσy 0 0 iσ y H mf (k) =PH mf ( k)p 1 [H mf (k)] T = T H mf ( k)t 1 P-inversion odd form factor high symmetry points H mf (k m )= 1 2 (ξ k m ε f )1 1 2 (ξ k m ε f )P Z 2 invariants are characterized by the parity eigenvalues: δ m = sgn(ξ k m ε f )

17 Topological Kondo insulators primitive unit cell (EASY!)! Z 2 invariants: 1 strong I STI = #')(" '(*(%(!,&%!"#$%&'() k z 8 δ m = ±1 m=1 3 weak I j WTI = δ m = ±1 k m P j phase diagram : tight binding 67&8 #')(" '(*(%(!,&% '(*(%(!,&%!"#$%&'()!"#$%&'() k z k z δ m = sgn(ξ k m ε f ) ξ k = 2t(cos cos cos k z )!"# !5 -./ :4!5!$# $# "#! !5 jour-ref: Phys. Rev. Lett. 104, (2010)!

18 Topological Kondo insulators bcc unit cell! Z 2 invariants: 1 strong I STI =( 1) w P j w P j 3 weak I (j) WTI =( 1)w P j phase diagram does not depend on the underlying type of the unit cell!"# #')(" '(*(%(!,&%!"#$%&'() k z !5 67&8 #')(" '(*(%(!,&% '(*(%(!,&%!"#$%&'()!"#$%&'() k z k z -./ :4!5!$# $# "#! !5!

19 Are existing Kondo insulators weak or strong TI? SmB 6 #')(" '(*(%(!,&%!"#$%&'() 67&8 '(*(%(!,&%!"#$%&'() #')(" '(*(%(!,&%!"#$%&'() V. N. Antonov et al. PRB (2006)! k z k z k z!"# !5 -./ :4!5!$# $# "#! !5!

20 Are existing Kondo insulators weak or strong TI? Ce-based Kondo insulators: localized moment n f =1 Nodes in the gap (CeNiSn)? CeNiSn CeRhSb Ce 3 Bi 4 Pt 3 J. N. Chazalviel et al. PRB (1976)! T. Terashima et al. PRB (2002)! #')(" '(*(%(!,&%!"#$%&'() k z 67&8 '(*(%(!,&%!"#$%&'() k z #')(" '(*(%(!,&%!"#$%&'() k z!"# !5 -./ :4!5!$# $# "#! !5!

21 CeNiSn: weak topological insulator? T. Takabatake et al. PRB (1990)! poly Depletion! in the! density of! states below! 10 K! Formation of! heavy fermions! polycrystalline samples: insulators single crystal samples: metallic T. Terashima et al. PRB (2002)!

22 Summary Ce-based Kondo insulators are weak topological insulators: unstable to disorder Strong Topological Insulators in f-electron systems are most likely to be observed in mixed-valence (n f 30%) materials Strong spin-orbit coupling due to hybridization of conduction electrons with f-electrons We expect fundamentally novel type of metallic states on the surface of the STI ω k µ ε f