Berry phase mediated Anomalous Thermoelectric and magnetic response - PowerPoint PPT Presentation

Berry phase mediated Anomalous Thermoelectric and magnetic response in 2D Topological Insulators Panagiotis Kotetes Institut f¨ ur Theoretische Festk¨ orperphysik, Karlsruhe Institute of Technology G. Sch¨ on (Karlsruhe Institute of Technology) A. Shnirman (Karlsruhe Institute of Technology) G. Varelogiannis (National Technical University of Athens) NanoCTM meeting: Balaton Hungary, June 16, 2011 Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Making a long story, short... Brief introduction to 2D Topological Insulators Sleuthing for unique fingerprints of Topology The emergence of an Anomalous Nernst effect and Orbital Magnetization Original Motivation: numerous experimental observations of a Giant Nernst signal in strongly correlated electronic systems The conditions for a Giant Nernst Signal in Chiral states of matter Anomalous thermoelectricity and magnetic response in planar Topological Semiconductors Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

2D Topological Insulators and bulk-boundary correspondence Definition of a Topological Insulator A state characterized by Topologically protected edge modes 2 fundamental systems : Anomalous Quantum Spin Hall Insulator → Time-Reversal Anomalous Quantum Hall Insulator → No Time-Reversal X.-L. Qi and S.-C. Zhang, Physics Today Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

2D T-invariant Topological HgTe-QW Semiconductors The Hamiltonian must include the following features Spin-Orbit coupling Band Gap near the Γ point Time-reversal symmetry Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

BHZ Quantum Spin Hall Insulator lattice model B HZ lattice model Hamiltonian:   0 , m s = + 1 | k , m l = 2 > � � � �  | k , m l = +1 , m s = + 1  H ( k ) 0 2 >   H ( k ) = ,  0 , m s = − 1  � � H ∗ ( − k ) | k , m l = 2 > 0 | k , m l = − 1 , m s = − 1 2 > with � H ( k ) = ε ( k ) + g ( k ) · τ where ε ( k ) = C − 2 D (2 − cos k x − cos k y ) � � �� − M g ( k ) = A sin k x , A sin k y , − 2 B 2 B + 2 − cos k x − cos k y Band Gap at the Γ point 2 E ( 0 ) ≡ 2 | g ( 0 ) | = 2 | M | B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry Curvature in k -space and Topological charge For each block, the avoided band touching at the Γ point Energy Bands X M � � generates a finite Berry Curvature for each band in a block � ∂ ˆ � ν ( k ) = − ν g ( k ) × ∂ ˆ g ( k ) Ω z 2 ˆ g ( k ) · ν = ± ∂ k x ∂ k y The Topological (Monopole) Charge that sources the Berry Curvature is equal to � N = − 1 � d 2 k Ω z − ( k ) = 1 2 π and provides the # of protected edge modes per block! Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry phase effects on thermoelectric transport and magnetic response The finite Berry curvature acts as a k − dependent magnetic field leading to Anomalous Charge Hall effect with Hall conductivity � σ xy = − e 2 1 Ω z ν ( k ) n F [ E ν ( k )] � N k ,ν For T = 0 and µ = 0 σ xy = − n e 2 h , n = 1 Anomalous thermoelectric effect with Hall conductivity � � 1 + e − β E ν ( k ) �� � e Ω z α xy = ν ( k ) E ν ( k ) n F [ E ν ( k )] + k B T ln T � N k ,ν Finite Orbital Magnetization � � 1 + e − β E ν ( k ) �� � M orb = e Ω z ν ( k ) E ( k ) n F [ E ν ( k )] + k B T ln � N k ,ν Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Berry-Curvature-originating fingerprints, for detecting Topological Order in 2D Anomalous Quantum Hall state: Single block Hamiltonian ! Anomalous Hall effect Anomalous charge Thermoelectric effect Finite Orbital Magnetization Anomalous Quantum Spin Hall state: Two block Hamiltonian with opposite Berry curvature per block ! Anomalous Spin Hall effect Anomalous Spin Thermoelectric effect Finite additional Zeeman Magnetization due to Orbital effects! Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Thermoelectric Transport and Nernst Signal Constitutive relations for thermoelectric charge transport J x = σ xx E x + σ xy E y + α xx ( − ∂ x T ) = σ yx E x + σ yy E y + α yx ( − ∂ x T ) J y with J ≡ charge current, E ≡ electric field, T ≡ temperature Thermopower S ⇒ longitudinal voltage appearing for J = 0 Nernst signal N ⇒ transverse voltage appearing for J = 0 E x S ≡ and ∂ x T E y N ≡ − ∂ x T Anomalous N → B z = 0 Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Quasiparticle and Vortex sources of a Nernst signal Quasiparticles Transverse velocity due to the Lorentz force ⇒ N ∼ B z 1 Nernst signal takes both signs depending on Doping 2 Nernst signal strongly linear in Temperature 3 Single band metals show a tiny Nernst signal ∼ nV / K 4 due to Sonheimer cancellation E. H. Sondheimer, Proc. R. Soc. London, Ser. A 193 , 484 (1948) Superconducting Vortices Normal Core Entropy + Vortex attached Flux ⇒ α xy � = 0 1 ⇒ N ∼ B z , B. D. Josephson, Physics Letters 16 , 242 (1965) Only Positive Nernst signal !!!! 2 Nernst signal non-linear in Temperature 3 Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Chirality driven Nernst signal Chirality ≡ Finite Angular Momentum Violation of Time-Reversal ⇒ σ xy ( B z = 0 ) � = 0 and 1 α xy ( B z = 0 ) � = 0 ⇒ Anomalous Hall + Nernst Effects! “Magnetic-field” in k-space: the Berry curvature Ω z ( k ) . 2 The Nernst signal takes both signs !!!! 3 Large Fermi-Surface ⇒ N linear in Temperature 4 Examples CuCr 2 Se 4 − x Br x : Spinel Ferromagnet + Spin-Orbit coupling Wei-Li Lee, S. Watauchi, V. L. Miller, R. J. Cava and N. P. Ong, Science 303 , 1647 (2004) Heavily-Doped Chiral d xy + id x 2 − y 2 Density Wave C. Zhang, S. Tewari, V. M. Yakovenko and S. Das Sarma, Phys. Rev. B 78 , 174508 (2008) But what happens in the Strongly Insulating limit ????? Chirality Induced Tilted-Hill Giant Nernst Signal: PK and G. Varelogiannis, PRL 104, 106404 (2010) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Giant Tilted-Hill Nernst signal in High-Tc cuprates Giant N in Pseudogap + Superconducting regimes Tilted-Hill (peaked) temperature profile Positive Nernst signal Enhanced Diamagnetism in the pseudogap phase Diamagnetism scales with the Nernst signal Yayu Wang, Lu Li, and N. P. Ong, PRB 73 , 024510 (2006) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Giant Tilted-Hill Nernst signal in the heavy fermion compound URu 2 Si 2 The non-SC order in the phase diagram ≡ “Hidden Order” (HO) Giant N in the Hidden Order Tilted-Hill temperature profile No Diamagnetism! For low T , the HO condenses in a SC state , possibly Topological R. Bel, H. Jin, K. Behnia, J. Flouquet and P. Lejay, PRB 70 , 220501 (2004) Y. S. Oh et al , PRL 98 , 016401 (2007) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Chiral d xy + id x 2 − y 2 Density Wave Chiral D-Density waves have been recently proposed for understanding the Pseudogap regime in the cuprates (PK and G. Varelogiannis 2008 & S. Tewari et al. 2008) and the Hidden Order (PK, A. Aperis and G. Varelogiannis 2010) The very-same interactions promoting unconventional superconductivity, also favour Chiral Density Wave formation Half-filled single band square lattice model: � (cos k x + cos k y ) c † H 0 = − 2 t k c k k Enhanced tendency towards an Insulating Chiral D-Density Wave due to perfect nesting Formation of a Topological Insulating Condensate Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Pairing interactions and Mean-field decoupling Intersite extended Hubbard interactions up to n.n.n. � � � V ij n i n j + J ij � S i · � H int = S j << i , j >> � k , k ′ V k , k ′ c † k c k + Q c † Driving effective interaction ∼ k ′ + Q c k ′ Chiral d-density wave “Anomalous” Terms ∆( k ) c † k c k + Q + h . c . Chiral D-Density Wave Order Parameter: ∆( k ) ∼ � k ′ V k , k ′ < c † k ′ + Q c k ′ > ⇒ ∆( k ) = ∆ 1 sin k x sin k y − i ∆ 2 (cos k x − cos k y ) Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators

Mean-field Hamiltonian of a chiral d-density wave Nambu isospinor Ψ † k = ( c † k c † k + Q ), k ∈ reduced B.Z. We obtain a pseudospin- 1 2 system for each k -point � � ε ( k ) + g 3 ( k ) g 1 ( k ) − ig 2 ( k ) H ( k ) = = ε ( k ) I τ + g ( k ) · τ g 1 ( k ) + ig 2 ( k ) ε ( k ) − g 3 ( k ) g 1 ( k ) = ∆ 1 sin k x sin k y , g 2 ( k ) = ∆ 2 (cos k x − cos k y ), g 3 ( k ) = − 2 t (cos k x + cos k y ) and ε ( k ) = − µ . 2-Band Energy Spectrum: ν = ± → E ν ( k ) = ε ( k ) + ν | g ( k ) | Karlsruhe Institute of Technology Panagiotis Kotetes Berry phase effects in 2D Topological Insulators