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How to determine the Hubbard U in materials containing transition metals

How to determine the Hubbard U in materials containing transition metals. Graduate school of Osaka Univ. Yoshida Lab. Atsushi Seike. Reference: M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71 (2005) 035105. Contents. Introduction Transition metal oxide Necessity of LDA+U

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How to determine the Hubbard U in materials containing transition metals

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  1. How to determine the Hubbard Uin materials containing transition metals Graduate school of Osaka Univ. Yoshida Lab. Atsushi Seike Reference: M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71 (2005) 035105.

  2. Contents • Introduction • Transition metal oxide • Necessity of LDA+U • Cococcioni-Gironcoli’s LDA+U • How to determine U ? • Result: calculations of Fe • Dependence of U on states and/or conditions of Fe • A new method to determine U • Summary

  3. Transition metal oxide (TMO) • Various physical properties in TMO • High-Tc superconductors: (La1-xSrx)2CuO4, etc. • Dielectric materials: BaTiO3 • Magnetic materials: Fe3O4 • Applications • Magneto resistive devices: Fe/MgO/Fe • Catalysts: La(Fe0.57Co0.38Pd0.05)O3 So, we desire to ‘design favorable materials by a theoretical approach.’

  4. Difficulty in the materials design of TMO • LDA & GGA are applied to designing materials. • Ex: spin-GGA • Fe/MgO/Fe as • TMRdevices • But for transition metal oxides this method seems to fail. • Ex：La2CuO4 The origin of the failure is missing of particle nature of electrons in the mean-field approach!

  5. Problems in the LDA method A band structure of La2CuO4 by LDA • LDA asserts that this material has metallic property. • But La2CuO4 is a Mott-insulator with an anti-ferromagnetic order at low temperatures. EF M. M. Korshunov et. al. Phys. Rev. B 72(2005) 165104

  6. What LDA+Ucan correct? Occupation number of a localized orbital in LDA and exact calculations. With U, we can correct this LDA error. Coulomb repulsiveinteraction ↑ Matteo Cococcioni and Stefano de GironcoliPhy. Rev. B 71(2005) 035105

  7. The target of this study • Focus on the Cococcioni-Giloncoli method to determine the Hubbard U. • Analyzing application to Fe, acquire the property of this method. • Propose a new method.

  8. How to determine“U”? Assume that the true energy functional contains the next contribution. Introduce a Lagrange multiplier. Use, We have,

  9. How to configure the response matrixes To obtain the response matrix, we apply a local potential shift by a at the site 1. From the center site, electrons flow out and go into the other sites. 1 According to the periodic boundary condition and translation symmetry, we can obtain the response matrix elements completely only with applying a to the particular transition metal site. 1 2 1 As a simple example, let’s think about 2-dimenssional crystal like left. (The region surrounded with blue line is the unit cell.) 3 4 3 1 2 1

  10. Determination of U on ferromagnetic a-Fe The crystal structure : bcc Lattice constant : a=5.217[a.u.] For the ferromagnetic iron, U is known to be a finite value. (Coccocioni-Gironcoli) Use Fe as a test ground!

  11. Calculation condition and method (Fe) Program Quantum espresso 3.2.3 Potential Ultra soft pseudo pontential 3s, 3p, 3d, 4s Exc : PBE Relativistic correction Cutoff Energy Wave function:35Ry Charge density:420Ry

  12. Results n v.s. a for Fe • We have several SCF solutions : • Ferromagnetic GS with a finite magnetic moment. • Weak ferromagnetic states with a small magnetic moment. • Non-magnetic states with no magnetic moment. • Energy difference between these states reproduces known results. • Calculated U=1.68[eV] for F-GS. The tangent of these curves gives U. ΔE=Enomag-Emag≒0.04[Ry/atom]

  13. Different results of calculated Uderived from different state of Fe or N The value of calculated U can vary because of the difference of magnetic state and/or N.

  14. A new method to determine Hubbard U: An extension of the CG method We can replace the LDA calculation by LDA+U in the determination process of U. • Do a first LDA+U calc. with a preset Upre. • Using the Cococcioni-Gironcoli method (CG), obtain a calculated Ucalc. • Continue calculation until Upre equal to Ucalc. Ucalc Upre

  15. Different results of calculated Uderived from different conditions It seems that calculated U tend to be small with magnetic ordering. Different preset U gives the different value of calculated U

  16. Summery • We have proposed a new method to determine ‘U’ with a parameter ‘preset Upre’. • We have shown that the method gives a self-consistently determined U. • The method can be applied to variousmaterials containing transition metals.

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