LDA band structures of transition-metal oxides Is it really the prototype Mott transition? The metal-insulator transition in V 2 O 3 and what electronic correlations may do to them Lecture 2.2, XV Training Course in the Physics of Strongly Correlated Systems, IASS Vietri sul Mare

[1] T. Saha-Dasgupta, O.K. Andersen, J. Nuss, A.I. Poteryaev, A. Georges, A.I. Lichtenstein; arXiv: [2] A.I. Poteryaev, J.M. Tomczak, S. Biermann, A. Georges, A.I. Lichtenstein, A.N. Rubtsov, T. Saha-Dasgupta, O.K. Andersen; Phys. Rev. B 76, (2007) [3] F. Rodolakis, P. Hansmann, J.-P. Rueff, A. Toschi, M.W. Haverkort, G. Sangiovanni, A. Tanaka, T. Saha-Dasgupta, O.K. Andersen, K. Held, M. Sikora, I. Alliot, J.-P. Itié, F. Baudelet, P.Wzietek, P. Metcalf, M. Marsi; Phys. Rev. Lett. 104, (2010). N. Parragh [4] S. Lupi, L. Baldassarre, B. Mansart, A. Perucchi, A. Barinov, P. Dudin, E. Papalazarou, F. Rodolakis, J.-P. Rueff, J.-P. Itié, S. Ravy, D. Nicoletti, P. Postorino, G. Sangiovanni, A. Toschi, P. Hansmann, N. Parragh, T. Saha-Dasgupta, O.K. Andersen, K. Held, M. Marsi; (accepted)

Doped Mott Insulators have rich physical properties and controlling them is one of the major challenges for developing Advanced Materials High-Temperature Superconductors Colossal Magneto-Resistance Materials Intelligent Windows, Field-effect Transistors

Hubbard model LDA+DMFT 1/2 fillingT=2000K, U = 3.0 eVU =2.1 eV T. Saha-Dasgupta and OKA 2002 Wannier orbital Conduction band (LDA)

U/W = 1 U/W = 2 U/W = 2.5 U/W = 3 U/W = 4 Georges and Kotliar 1992: The single-band Hubbard Model in the d=∞ limit can be mapped exactly onto the Anderson impurity model supplemented by a CPA-like self- consistency condition for the dynamical coupling to the non- interacting medium. Hence, the Kondo- resonance may develop into a quasi- particle peak. For general hopping, the Georges-Kotliar mapping leads to the dynamical mean-field approximation(DMFT). A. Georges et al, Rev Mod Phys 1996: QP Gap LHBUHB W = 1 LDA O.K. LDA+U O.K. DMFT needed Mott transition

Electronic-structure calculations for materials with strong correlations Current approximations to ab inito Density-Functional Theory (LDA) are insufficient for conduction bands with strong electronic correlations, e.g. they do not account for the Mott metal-insulator transition. On the other hand, LDA Fermi surfaces are accurate for most metals, including overdoped high-temperature superconductors. Presently, we therefore start with the LDA. For the few correlated bands, we then construct localized Wannier orbitals (NMTOs) and a corresponding low-energy Hubbard Hamiltonian: H LDA + U on-site. This is solved in the dynamical mean-field approximation (DMFT).

V 3d 2 AFI monoclinic Paramagnetic M and I corundum str LDA+U: Ezhov, Anisimov, Khomskii, Sawatzky 1999 M I M AFI I M

LDA band structure of V 2 O 3 projected onto various orbital characters: N=2 N=1 Blow up the energy scale and split the panels: EFEFEFEF EFEFEFEF EFEFEFEF Pick various sub- bands by generating the corresponding minimal NMTO basis set: EFEFEFEF For the low-energy Hamiltonian we just need the t 2g set

(V 1-x M x ) 2 O 3 V 2 O 3 3d (t 2g ) 2 Hund's-rule coupling J=0.7 eV a 1g -e g π crystal-field splitting = 0.3 eV

Undo hybridization a 1g egπegπegπegπ PM LDA t 2g NMTO Wannier Hamiltonian 2.0 LDA PM a 1g -e g π crystal-field splitting = 0.3 eV U-enhancement = 1.85 eV ~ 3J LDA+DMFT U = 4.25 eV, J = 0.7 eV Crystal-field enhanced and mass-renormalized QP bands PM 390 K

Comparison with PES (Mo et al. PRL 2004):

PM e g electrons are "localized" and only coherent below ~250K "itinerant" and coherent below ~400K a 1g electrons are "itinerant" and coherent below ~400K More important for the temperature dependence of the conductivity is, however, that internal structural parameters of V 2 O 3 change with temperature, as we shall see later.

Undo hybridization a 1g egπegπegπegπ PM LDA t 2g NMTO Wannier Hamiltonian 2.0 LDA PM a 1g -e g π crystal-field splitting = 0.3 eV U-enhancement = 1.85 eV ~ 3J LDA+DMFT U = 4.25 eV, J = 0.7 eV Crystal-field enhanced and mass-renormalized QP bands PM egπegπegπegπ a 1g PI = − Undo hybridization a 1g egπegπegπegπ 390 K

U=4.2 eV, 3.8% Cr, T=580 KU=4.2 eV, 0 % Cr, T=390 K PMPI

t = eV

t = eV

t = eV t = eV

2.0 eV V2O3V2O3 T=300K (V 0.96 Cr 0.04 ) 2 O 3 LDA LDA LDA LDA undo a 1g -e g π (V 0.99 Cr 0.01 ) 2 O 3 Robinson, Acta Cryst. 1975: (V 0.99 Cr 0.01 ) 2 O 3 V 2 O 3 at 300K ~ ~ V 2 O 3 at 900K

V 2 O 3 3d (t 2g ) 2 Hund's-rule coupling

not, This metal-insulator transition in V 2 O 3 is not, like in the case of a single band, e.g. a HTSC: Hubbard model, LDA+DMFT Band 1/2 full U = 3.0 eV T=2000K Wannier orbital and LDA conduction band U =2.1 eV T. Saha-Dasgupta and OKA 2002 it is not really a Mott transition. caused by disappearance of the quasi-particle peak and driven by the Coulomb repulsion (U), i.e. it is not really a Mott transition.

Conclusion t 2g correlation-enhanced a 1g -e g π crystal-field splitting and lattice distortion In the (t 2g ) 2 system V 2 O 3, described by an LDA t 2g Hubbard model, the metal-insulator transition calculated in the DMFT is caused by quasi-particle bands being separated by correlation-enhanced a 1g -e g π crystal-field splitting and lattice distortion. The driving mechanism is multiplet splitting (nJ) rather than direct Coulomb repulsion (U). a 1g coherent e g π The a 1g electrons stay coherent to higher temperatures (~450K) than the e g π electrons (~250K).