First-Principles Calculations for Multiferroic BiFeO3

Contents 10.1 Introduction to First-Principles Calculations ........................................ 281 10.2 Applications of the Ab Initio Method to Multiferroic Materials ......... 285 10.2.1 Ferroelectric, Magnetic, and Electronic Properties of BiFeO3 in Bulk R3c and Strained P4mm Structures ................ 285 10.2.2 Heterostructures of BiFeO3 with Other Transition Metal Perovskites ...................................................................................... 293 10.3 Conclusions and Outlook .......................................................................... 295 Acknowledgments ................................................................................................. 296 References ............................................................................................................... 296 In Equation 10.1, the rst three terms represent electron kinetic energy, the Coulomb potential of electrons in a nuclear environment, and electronelectron Coulomb repulsion, while the last two terms represent the nucleus kinetic energy and the nucleon-nucleon Coulomb repulsion. Here, me and MI are the masses of individual electrons and nuclei; the symbol -e stands for the electron charge and ZIe denotes the nucleus charge; ri and RI are the coordinates of electronic and nuclear degrees of freedom. Since the Hamiltonian involves enormous degrees of freedom, it is forbiddingly di cult to solve the entire problem quantum mechanically and approximations must be introduced. In the Oppenheimer approximation, because the mass of a nucleus is much larger than that of an electron, the nuclei move much slower than the electrons such that one can focus on the electronic degrees of freedom described by the Schrödinger equation: ∑ ∑ ∑= − ∇ − − + − H m Z e r R e r rei 1 2 1 2el (10.2) as if the nuclei are xed in motion. e motion of the latter will be treated classically in an eective medium with electronic contribution through the Feynman-Hellman theorem. Even so, the electronic degrees of freedom are still highly entangled and it is a challenging problem to solve the electronicrelated Schrödinger Equation 10.2. Within the density functional theory (DFT), the original many-electron problem can be mapped onto a dierent auxiliary system. In the Kohn-Sham ansatz, the ground state density of the original interacting electron system is equal to a much simplied Kohn-Sham particle system [1]. e motion of these Kohn-Sham particles is described by the single-particle Schrödinger equation: m r r r e2 eff ( ) ( ) ( )− ∇ + ν   ψ = ε ψνσ ν νσ (10.3) where ) ) ) )( ( ( (ν = ν + ν + νσr r r reff ext Hartree xc (10.4) consists of the external potential arising from the nuclei and any other external elds )(ν rext , the Hartree potential r n rr r rdHartree ∫ ( )( )ν = − ′ ′, and the exchange- correlation potential )(ν rxc . Here the total electron density is in turn given by the solution to the Eigen-Equation 10.3: n r n r r 1 ∑ ∑∑) ) )( ( (= = ψσ for the total N = N↑+N↓ electrons occupying N lowest orbitals. As such, the above Kohn-Sham equation should be solved self-consistently. We note that the external applied eld can also be spin dependent when the spin Zeeman interaction in the presence of an applied magnetic eld becomes important in some situations. In the Kohn-Sham ansatz, the crucial quantity is the exchange-correlation energy, the functional derivative of which determines the exchange-correlation potential )(νσ rxc . Currently, the two most widely used potentials are based on the local density approximation (LDA) [2] and the generalized gradient approximation (GGA) [3]. For the past several decades, the LDA or GGA in the DFT framework has been very successful in describing structural, electronic, and optical properties of many good metals and several semiconductors, where the electronic correlations are rather weak. However, when the LDA-based DFT method is applied to complex materials like transition metal oxides (TMOs) and heavy-fermion materials with open d or f shell electrons, the description is not adequate. For example, TMOs like MnO, NiO, La2CuO4 are known Mott insulators, while the LDA predicts that they are metals. ere are several attempts to improve the LDA approach including the self-interaction correction (SIC) method [4], the GW approximation [5,6], the hybrid DFT [7], the LDA + U method [8], and the more recently developed combination of LDA with dynamical mean-eld theory (DMFT), the so-called LDA + DMFT method [9,10]. We note that the SIC method usually gives incorrect location of occupied d-bands in these TMOs and is also numerically expensive, while the GW method in its standard implementation works reasonably well for weakly interacting semiconductors. e LDA + DMFT method is a truly quantum many-body approach, which captures dynamical correlation eects and the localization-delocalization transition in strongly correlated electron systems. When one is in particular interested in the electronic and magnetic behaviors in the strongly correlated insulators, the LDA + U method turns out to be appealing with its computational e - ciency and relatively transparent physical interpretation. e LDA + U total energy functional has proven to give large but reasonable corrections for a number of magnetic insulators. erefore, we describe this method here with more technical details.