LCI -birthplace of liquid crystal display. May, protests. Fashion school is in top-3 in USA. Clinical Psychology program is Top-5 in USA
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1 LCI -birthplace of liquid crystal display May, protests Fashion school is in top-3 in USA Clinical Psychology program is Top-5 in USA
2 Topological insulators driven by electron spin Maxim Dzero Kent State University (USA) and CFIF, InstitutoSuperior Tecnico(Portugal)
3 Collaborators: Kai Sun, U. Michigan Piers Coleman, Rutgers U. Victor Galitski, U. of Maryland Victor Alexandrov, CUNY Victor A. Victor G. Kai Bitan Roy, U. of Maryland Jay Deep Sau, U. of Maryland Maxim Vavilov, U. Wisconsin-M References: Jay Deep Piers Maxim MD, K. Sun, V. Galitski& P. Coleman, Phys. Rev. Lett. 104, (2010) MD, K. Sun, P. Coleman & V. Galitski, Phys. Rev. B 85, (2012) MD, Europhys. Jour. B 85, 297 (2012) V. Alexandrov, MD & P. Coleman, Phys. Rev. Lett. 111, (2013) B. Roy, Jay Deep Sau, MD & V. Galitski, arxiv: (2014) MD, M. Vavilov& V. Galitski, preprint (2014)
4 Model Hamiltonian Special case will be discussed:
5 Quest for topological insulators beyond Bi-based materials complex materials with d-& f-orbitals increasing localization increasing localization Coleman (2002) A lot of action takes place on the brink of localization!
6 Quest for ideal topological insulators complex materials with d-& f-orbitals 3d & 4d-orbitals 5d-orbitals 4f-orbitals 5f-orbitals
7 Quest for ideal topological insulators complex materials with d-& f-orbitals candidates for f-orbital topological insulators FeSb 2, SmB 6, YbB 12, YbB 6 & Ce 3 Bi 4 Pt 3
8 f-orbital insulators: Anderson lattice model conduction electrons (s,p,d orbitals) f-electrons 2J+ 1 tetragonal crystal field 2J+ 1 cubic crystal field
9 f-orbital insulators: Anderson lattice model conduction electrons (s,p,d orbitals) f-electrons hybridization: matrix element Non-local hybridization odd functions of k Strong spin-orbit coupling is encoded in hybridization
10 Non-interacting limit Anderson lattice model: U=0 basis Hamiltonian (2D) c-f hybridization: Equivalent to Bernevig-Hughes-Zhang (BHZ) model
11 Non-interacting limit Anderson lattice model (2D): U=0 Jan Wernerand FakherF. Assaad, PRB 88, (2013)
12 Finite-U: local moment formation P. W. Anderson, Phys. Rev 124, 41,(1961) local d-or f-electron resonance splits to form a local moment
13 Heavy Fermion Primer: Kondo impurity Ceor Smimpurity total angular momentum J=5/2 Electron sea Pauli Spin (4f,5f): is screened basicby fabric conduction of heavy electrons electron physics 2J+ 1 Kondo Temperature Curie
14 Heavy Fermion Primer coherent heavy fermions Kondo lattice
15 Semiconductors with f-electrons canonical examples: SmB 6 & YbB 12 Ce 3 Bi 4 Pt 3 exponential growth M. Bat kova et. al, Physica B , 618 (2006) Conductivity remains finite! P. Canfield et. al, (2003)
16 Mott s Hybridization picture N. Mott, Phil. Mag. 30, 403 (1974) Formation of Heavy f-bands: electrons and localizedfdoublets hybridize, possibly due to Kondo effect Mott, 1973
17 Mott s Hybridization picture N. Mott, Phil. Mag. 30, 403 (1974) Formation of heavy-fermion insulator n c +n f =2 integer canonical examples: SmB 6 & YbB 12
18 Interacting electrons: Kondo insulators Anderson model: infinite-u limit Projection (slave-boson) operators: Constraint: infinite-u limit: hamiltonian mean-field approximation:
19 Kondo insulators: mean-field theory self-consistency equations: mean-field Hamiltonian: renormalized position of the f-level
20 Kondo insulators: mean-field theory mean-field Hamiltonian: L. Fu & C. Kane, PRB 76, (2007) parity renormalized position of the f-level time-reversal P-inversion odd form factor high symmetry points of the Brillouin zone
21 Tetragonal Topological Kondo Insulators Z 2 invariants: strong : 3 weak : tetragonal symmetry: Kramers doublet BI STI WTI Strong mixed valence favors strong topological insulator! jour-ref: Phys. Rev. Lett. 104, (2010)
22 Strong Topological Kondo Insulators BI STI WTI Q: What factors are important for extending strong topological insulating state to the local moment regime(n f 1)? A: Degeneracy = high symmetry (cubic!)
23 Topological Kondo Insulators: Large-N theory Replicate Kramersdoublet Ntimes: SU(2) -> SP(2N) SU(2) -> SP(2N): time-reversal symmetry is preserved control parameter: 1/2N M. Dzero, Europhys. Jour. B 85, 297 (2012)
24 Cubic Topological Kondo Insulators Cubic symmetry (quartet): SmB 6 T. Takimoto, Jour. Phys. Soc. Jpn. 80, (2011) V. Alexandrov, M. Dzero, P. Coleman, Phys. Rev. Lett. 111, (2013) Antonov,Harmon,Yaresko, PRB (2006) 4d 5f V. Alexandrov, M. Dzero, P. Coleman, Phys. Rev. Lett. 111, (2013) Bands must invert X or M high symmetry points
25 Cubic Topological Kondo Insulators Cubic symmetry (quartet): SmB 6 4d 5f BI STI V. Alexandrov, M. Dzero, P. Coleman, Phys. Rev. Lett. 111, (2013)) Cubic symmetry protects strong topological insulator!
26 Mean-field theory for SmB 6 : is N=1/4 small enough? integrated spectral weight of the gap: T-dependence full insulating gap: dependence on pressure Derret al. PRB 77, (2008) Nyhus, Cooper, Fisk, Sarrao, PRB 55, (1997) Mean-field-like onset of the insulating gap!
27 Cubic Topological Kondo Insulators: surface states Bulk Hamiltonian: Assumption: boundary has little effect on the bulk parameters, i.e. mean-field theory in the bulk still holds with open boundaries
28 Cubic Topological Kondo Insulators: surface states effective surface Hamiltonian: Fermi velocities are small: surface electrons are heavy
29 Cubic Topological Kondo Insulators: surface states three Dirac cones: Gamma point, other X points no surface potential Surface potential has two effects: band bending: surface carriers are light! Dirac point moves into the valence band: confirmed by ARPES experiments!
30 What makes topological Kondo insulators special? Q: Are topological Kondo insulators adiabatically connected to topological band insulators? A: NO! Gap closes as the strength of U gradually increases Jan Wernerand FakherF. Assaad, PRB 88, (2013)
31 SmB 6 : potential candidate for correlated TI
32 SmB 6 : experiments Q: Can we establish that SmB 6 hosts helical surface states with Dirac spectrum while relying on experimental data only? (1) transport is limited to the surface (2) time-reversal symmetry breaking leads to localization (3) strong spin-orbit coupling = helicity (4) Dirac spectrum
33 Idea: pass the current and measure voltage drop surface transport bulk transport S. Wolgastet al., PRB 88, T < 5K transport comes from the surface ONLY!
34 Idea: Ohm s law in ideal topo insultor resistivity is independent of sample s thickness: surface transport Resistivity ratio Thickness independent! D. J. Kim, J. Xia & Z. Fisk, Nat. Comm. (2014)
35 SmB 6 : transport experiments Non-magnetic ions (Y) on the surface: time-reversal symmetry is preserved X e e No backscattering Magnetic ions (Gd) on the surface: time-reversal symmetry is broken e Localization e Kim, Xia & Fisk, arxiv:
36 SmB 6 : quantum oscillations experiments Idea:zero-energy Landau level exists for Dirac electrons: Experimental consequence: shift in Landau index nmust be observed only E 0 =0 contributes at infinite magnetic field! very light effective mass: m e Strong surface potential! G. Li et al. (Li Lu group Ann Arbor) arxiv:
37 SmB 6 : weak anti-localization (WAL) Weak SO coupling -> WL Strong SO coupling -> WAL
38 SmB 6 : weak anti-localization *diffusion (classical) conductivity is field-independent S. Thomas et al. arxiv:
39 Conclusion & open questions Role of correlations? Topological Kondo insulators from artificial structures? [Can we increase the which only surface is conducting?] Why all Kondo insulators have cubic symmetry? [Kondo semimetals have tetragonal symmetry] Pressure- or chemical substitution-driven superconductivity? effect magnetic vs. non-magnetic doping on the magnitude of surface conductivity surface conductance & insulating bulk below 5K SmB 6 is a correlated topological insulator quantum oscillations experiments confirm Dirac Dirac spectrum of surface electrons weak anti-localization: strong SO coupling
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