Institute for Theoretical Physics Competing interactions and magnetic order in correlated electron systems Thomas Pruschke and Robert Peters Department of Theoretical Physics University of Göttingen Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 1
Institute for Theoretical Physics Acknowledgements Actual work done by Robert Peters Computer support: Gesellschaft für Wissenschaftliche Datenverarbeitung Göttingen Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen Financial support by: DFG through SFB 602 and Pr 298/10 Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 2
Institute for Theoretical Physics e g ∆ CF cubic environment U , U ′ , J H t 2 g S = 3 2 Mn 3+ Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 3
Institute for Theoretical Physics ∆ JT e g ∆ CF cubic environment U , U ′ , J H + Jahn-Teller t 2 g S = 3 2 Mn 3+ Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 3
Institute for Theoretical Physics Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 4
Institute for Theoretical Physics TMO like LaMnO 3 , LaTiO 3 , V 2 O 3 etc. • Correlated metals, ..., paramagnetic insulators • Different types of magnetic and orbital ordering • Competing interactions: High-spin/low-spin and magnetic frustration Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 4
Institute for Theoretical Physics TMO like LaMnO 3 , LaTiO 3 , V 2 O 3 etc. • Correlated metals, ..., paramagnetic insulators • Different types of magnetic and orbital ordering • Competing interactions: High-spin/low-spin and magnetic frustration GdN and other rare-earth monopnictides • Gd 3+ in GdN ⇒ localized Gd 4f 7 + Gd 5d conduction states ➡ localized Gd S=7/2 spin coupled ferromagnetically to conduction states • Band structure theory: Semiconductor • Experiment: Ferromagnetic semimetal Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 4
Institute for Theoretical Physics no f-electron contribution to band states Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 5
Institute for Theoretical Physics Heavy-fermion materials: Rare-earth and Actinides • Archetype: Compounds based on Ce ➡ localized S=1/2 spin • Band structure theory: Conventional metals no f-electron contribution to band states • Experimental findings: Strongly enhanced Fermi liquid parameters, several ordered phases • Well understood: Nature of heavy Fermi liquid ⇒ hybridization between 4f and band states • Open questions: Magnetism & supercondconductivity, quantum phase transitions Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 5
Institute for Theoretical Physics Heavy-fermion materials: Rare-earth and Actinides • Archetype: Compounds based on Ce ➡ localized S=1/2 spin • Band structure theory: Conventional metals no f-electron contribution to band states • Experimental findings: Strongly enhanced Fermi liquid parameters, several ordered phases • Well understood: Nature of heavy Fermi liquid ⇒ hybridization between 4f and band states • Open questions: Magnetism & supercondconductivity, quantum phase transitions Exotic „heavy-fermion“ material: LiV 2 O 4 • Heavy-fermion like features, origin heavily debated ‣ Magnetic frustration of corundum structure ‣ Partial localization of correlated 3d states + non-local hybridization effects Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 5
Institute for Theoretical Physics Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: H = t αβ ij, σ c † � � � i ασ c j βσ ➡ tight-binding d-like conduction states, σ ij αβ ε i ασ c † � including crystal-filed splittings + i ασ c i ασ i ασ Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: H = t αβ ij, σ c † � � � i ασ c j βσ ➡ tight-binding d-like conduction states, σ ij αβ ε i ασ c † � including crystal-filed splittings + i ασ c i ασ i ασ U αβγδ 2 c † i ασ 1 c † � � ➡ Coulomb interactions among conduction states i βσ 2 c i γσ ′ 2 c i δσ ′ + σ 1 σ 2 σ ′ 2 σ ′ 1 i αβγδ σ 1 σ ′ 1 σ 2 σ ′ 2 Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: H = t αβ ij, σ c † � � � i ασ c j βσ ➡ tight-binding d-like conduction states, σ ij αβ ε i ασ c † � including crystal-filed splittings + i ασ c i ασ i ασ U αβγδ 2 c † i ασ 1 c † � � ➡ Coulomb interactions among conduction states i βσ 2 c i γσ ′ 2 c i δσ ′ + σ 1 σ 2 σ ′ 2 σ ′ 1 i αβγδ σ 1 σ ′ 1 ➡ Localized spins σ 2 σ ′ 2 Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: H = t αβ ij, σ c † � � � i ασ c j βσ ➡ tight-binding d-like conduction states, σ ij αβ ε i ασ c † � including crystal-filed splittings + i ασ c i ασ i ασ U αβγδ 2 c † i ασ 1 c † � � ➡ Coulomb interactions among conduction states i βσ 2 c i γσ ′ 2 c i δσ ′ + σ 1 σ 2 σ ′ 2 σ ′ 1 i αβγδ σ 1 σ ′ 1 ➡ Localized spins σ 2 σ ′ 2 � � � � � J αβ σσ ′ c † � ➡ Exchange coupling between localized spins S i · � τ σσ ′ c i βσ ′ − i ασ i αβ σσ ′ and conduction states Origin: Hund‘s exchange (J>0) or Schrieffer-Wolff exchange (J<0) Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: H = t αβ ij, σ c † � � � i ασ c j βσ ➡ tight-binding d-like conduction states, σ ij αβ ε i ασ c † � including crystal-filed splittings + i ασ c i ασ i ασ U αβγδ 2 c † i ασ 1 c † � � ➡ Coulomb interactions among conduction states i βσ 2 c i γσ ′ 2 c i δσ ′ + σ 1 σ 2 σ ′ 2 σ ′ 1 i αβγδ σ 1 σ ′ 1 ➡ Localized spins σ 2 σ ′ 2 � � � � � J αβ σσ ′ c † � ➡ Exchange coupling between localized spins S i · � τ σσ ′ c i βσ ′ − i ασ i αβ σσ ′ and conduction states Origin: Hund‘s exchange (J>0) or Schrieffer-Wolff exchange (J<0) Hubbard-Kondo model Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 6
Institute for Theoretical Physics Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 7
Institute for Theoretical Physics Further simplifications: • Jahn-Teller or crystal field: Low T governed by single „d“ subband at Fermi level ➡ Conduction states modelled by single-band Hubbard model • Isotropic exchange to local „f“ spin � 2 � � � σ n ( d ) n ( d ) S ( f ) � � � � t ij d † ε ( d ) d † � H = i σ d j σ + i σ + U − J · � τ σσ ′ d i σ ′ − 1 i i σ i i,j, σ i σ i i σσ ′ • Large U ⇒ TMO 3d states, small U ⇒ rare-earth 5d states; J<0 ⇒ heavy fermions Competing interactions and magnetic order Concepts in Electron Correlations in correlated electron systems Hvar 2008 7
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