First-principles theory of orbital magnetization The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Ceresoli, Davide et al. “First-principles theory of orbital magnetization.” Physical Review B 81.6 (2010): 060409. © 2010 The American Physical Society As Published http://dx.doi.org/10.1103/PhysRevB.81.060409 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/56261 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. RAPID COMMUNICATIONS PHYSICAL REVIEW B 81, 060409͑R͒͑2010͒ First-principles theory of orbital magnetization Davide Ceresoli,1 Uwe Gerstmann,2 Ari P. Seitsonen,2 and Francesco Mauri2 1Department of Materials Science and Engineering, Massachusetts Institute of Technology (MIT), 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA 2IMPMC, CNRS, IPGP, Université Paris 6, Paris 7, 140 rue de Lourmel, F-75015 Paris, France ͑Received 3 February 2010; published 19 February 2010͒ Within density-functional theory we compute the orbital magnetization for periodic systems evaluating a recently discovered Berry-phase formula. For the ferromagnetic metals Fe, Co, and Ni we explicitly calculate the contribution of the interstitial regions neglected so far in literature. We also use the orbital magnetization to compute the electron paramagnetic resonance g tensor in paramagnetic systems. Here the method can also be applied in cases where linear-response theory fails, e.g., radicals and defects with an orbital-degenerate ground state or those containing heavy atoms. DOI: 10.1103/PhysRevB.81.060409 PACS number͑s͒: 75.20.Ϫg, 71.15.Ϫm, 76.30.Ϫv The electric polarization and the orbital magnetization are stitial regions contribute by up to 50% to the orbital mag- well-known textbook topics in electromagnetism and solid- netic moments. So far neglected in the literature these state physics. While it is easy to compute their derivatives in contributions are thus shown to be one source for underesti- an extended system, the electric polarization and the orbital mated ab initio values. Furthermore we make use of a rela- magnetization themselves are not easy to formulate in the tionship between the orbital magnetization and the electronic thermodynamic limit due to the unboundedness of the posi- g tensor that can be measured in electron paramagnetic reso- tion operator. The problem of the electric polarization has nance ͑EPR͒ experiments.8 We propose a nonperturbative been solved in the 1990s by the modern theory of polariza- method that is highly superior to existing linear-response ͑ ͒ 1,2 tion MTP , which relates the electric polarization to the ͑LR͒ approaches9,10 since it can deal with systems in which Berry phase of the electrons. A corresponding formula for 3,4 spin-orbit coupling cannot be described as a perturbation. the orbital magnetization has been found very recently The total ͑sum of spin and orbital͒ magnetization can be showing that this genuine bulk quantity can be evaluated defined from the derivative energy E with respect to the from the ground-state Bloch wave functions of the periodic tot magnetic field B system. Since the discovery of the MTP, a wealth of papers Hץ Eץ -have appeared reporting its successful applications to first M ϵͯ− totͯ = ͚ f ͗␺ ͉ − ͉␺ ͘ , ͑1͒ n B=0 ץ n n ץ principles calculations of dielectric and piezoelectric properties.2 On the other hand, ab initio calculations of the B B=0 n B orbital magnetization via the Berry phase formula have not where f is the occupation of the eigenstate n and in the most been reported in literature yet, except than for simple tight- n general case the expectation value is to be taken on ground- binding lattice models. state spinors ␺ . In the last equality we take advantage of the The origin of the orbital magnetization in molecules and n solids is time-reversal breaking caused by e.g., spin-orbit Hellmann-Feynman theorem. The Hamiltonian in atomic ͑SO͒ coupling. In ferromagnetic materials the orbital magne- units is tization is a not negligible contribution to the total magneti- 1 ␣2gЈ ,zation. Several papers in literature5,6 showed that the orbital H = ͓p + ␣A͑r͔͒2 + V͑r͒ + ␴ · ٌ͓V͑r͒ ϫ ͑p + ␣A͑r͔͒͒ 2 8 magnetic moment of simple ferromagnetic metals ͑Fe, Co, and Ni͒ is strongly underestimated within density-functional ͑2͒ theory ͑DFT͒ if using the local-density approximation ͑LDA͒ ͑ ͒ where we drop the trivial spin-Zeeman term, reducing the or generalized gradient approximations GGA . Empirical magnetization according Eq. ͑1͒ only to its orbital part. We corrections such as the orbital polarization ͑OP͒͑Ref. 7͒ 1 use the symmetric gauge A͑r͒= Bϫr. The last term in Eq. have been thus employed to obtain a better agreement with 2 ͑2͒ is the leading spin-orbit term, describing the on-site SO the experimental values. Nevertheless it remains an interest- coupling ͓with fine structure constant ␣=1/c and the abbre- ing question if, e.g., functionals beyond LDA or GGA would viation gЈ=2͑g −1͒͑Refs. 9 and 10͔͒ and ␴ are the Pauli be able to describe the orbital magnetization correctly.6 All e matrices. We neglect the spin other orbit ͑SOO͒ term, in previous ab initio calculations have been however carried general a small contribution to the orbital magnetization and out in the muffin-tin ͑MT͒ approximation, i.e., computing the to the g tensor.11 orbital magnetization only in a spherical region centered on By inserting Eq. ͑2͒ into Eq. ͑1͒ we obtain the atoms, neglecting the contribution of the interstitial re- gion. ␣ M = ͚ f ͗␺ ͉r ϫ v͉␺ ͘, ͑3͒ In this Rapid Communication, we present first-principles 2 n n n DFT calculations of the orbital magnetization by evaluating n the recently discovered Berry phase formula.3,4 For the fer- where v=−i͓r,H͔, with H and ␺ computed at B=0. This romagnetic phases of Fe, Co, and Ni we show that the inter- expression can be directly evaluated in a finite system but 1098-0121/2010/81͑6͒/060409͑4͒ 060409-1 ©2010 The American Physical Society RAPID COMMUNICATIONS CERESOLI et al. PHYSICAL REVIEW B 81, 060409͑R͒͑2010͒ not in extended systems because of the unboundedness of the ␣ 1 ⌬M = ͚ ͳ͑R − r͒ ϫ ͓r − R,VNL͔ʹ, ͑11͒ position operator and of the contribution of itinerant surface bare 2 i R currents.3 However, in periodic systems and in the thermo- R dynamic limit, Eq. ͑3͒ can rewritten as a bulk property:3,4 gЈ␣3 1 ␣N ⌬ ͚ ͳ͑ ͒ ϫ ͓ NL͔ʹ ͑ ͒ ͘ Mpara = R − r r − R,FR , 12 ץ͉͒ ⑀ ⑀ ϫ ͑H ͉ ץ͗ c ϫ M =− Im͚ fnk kunk k + nk −2 F kunk , 16 R i 2Nk nk ͑4͒ gЈ␣3 ⌬ ͗ NL͘ ͑ ͒ H ⑀ Mdia = ͚ ER , 13 where k is the crystal Hamiltonian with B=0, nk and unk 16 ⑀ R are its eigenvalues and eigenvectors, F is the Fermi level, Nc is the number of cells in the system, and N is the number of ͗ ͘ ͚ ͗ ͉ ͉ ͘ k where ... stands for nkfnk ¯unk ... ¯unk . k points. In a periodic system Mbare can be nicely calculated by Equations ͑3͒ and ͑4͒ are valid at an all-electron ͑AE͒ evaluating Eq. ͑4͒ for the GIPAW Hamiltonian H and corre- level. To compute the orbital magnetization within a pseudo- ⑀ sponding PS eigenvectors ¯unk and eigenvalues ¯nk. All the potential ͑PS͒ approach, we recall that a PS Hamiltonian ͑H͒ reconstruction terms ͓Eqs. ͑11͒–͑13͔͒ can be easily evaluated reproduces by construction differences and derivatives of the in extended systems since the nonlocal operators VNL, FNL ͑ ͒ R R total energy. Thus we can still obtain M, from Eq. 1 ,ifwe and ENL act only inside finite spherical regions, centered ␺ R ץ/Hץ replace B and n by the corresponding PS quantities around each atom. ␺¯ ץ/Hץ B and n. The approach presented so far allows the calculation of We obtain the PS Hamiltonian in presence of spin-orbit the orbital magnetization in a general PS scheme including coupling and uniform magnetic field with the gauge includ- noncollinear spin polarization. In this work for the sake of ing projector augmented waves ͑GIPAW͒ method.12 In par- simplicity we use a collinear implementation. All expectation H T+HT H ͑ ͒ T ticular = B B, where is given by Eq. 2 and B is the values are evaluated by assuming decoupled spin channels GIPAW transformation ͓Eq. ͑16͒ of Ref. 12͔. If the AE and along the spin direction e. In particular all the spinors are PS partial waves have the same norm the GIPAW Hamil- eigenvectors of ␴·e and the local and total spin ͑S=Se͒ are tonian is given by aligned along e. Since the choice of e changes the spin-orbit coupling, the orbital magnetization is a function of e. In fer- H H͑0͒ H͑0͒ H͑1͒ H͑1͒ ͑ 2͒ = + SO + + SO + O B , romagnets, each spin direction e is characterized by a corre- where sponding total energy, whereby the minimum of the total energy with respect to e defines the preferred direction of the 1 spin alignment, the so-called easy axis of the ferromagnet. H͑0͒ = p2 + V ͑r͒ + VNL, ͑5͒ 2 ps R We implemented our method in the QUANTUM-ESPRESSO plane-wave code.14 We use standard norm-conserving 15 gЈ pseudopotentials with two GIPAW projectors per angular ͒ ͑ ͒ ͑ H͑0͒ ␣2 ␴ ٌ͑ ͑ ͒ ϫ ͒ NL SO = ͫ · Vps r p + ͚ FR ͬ, 6 momentum channel.