First-Principles Theory of Orbital Magnetization

First-principles theory of orbital magnetization

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As Published http://dx.doi.org/10.1103/PhysRevB.81.060409

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Citable link http://hdl.handle.net/1721.1/56261

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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 region. ␣ 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

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. Using spin-polarized LDA Ref. 16 and 8 R PBE ͑Ref. 17͒ functionals, we perform standard self- consistent-field ͑SCF͒ calculations including the SO term of ␣ 1 Eq. ͑6͒ in the collinear approximation within the Hamil- H͑1͒ = B · ͩL + ͚ R ϫ ͓r,VNL͔ͪ, ͑7͒ 2 i R tonian. Then we evaluate the orbital magnetization, accord- R ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ing to the Eqs. 4 , 9 , and 11 – 13 for Mbare. We neglect any explicit dependence of the exchange-correlation func- ͑ ͒ gЈ H 1 = ␣3B · ͩr ϫ ͑␴ ϫ ٌV ͒ + ͚ ENL tional on the current density. In practice, spin-current SO ps R ͑ ͒ 16 R density-functional theory SCDFT calculations have shown to produce negligible corrections to the orbital 1 6 ͑ ͒ + ͚ R ϫ ͓r,FNL͔ͪ. ͑8͒ magnetization. We compute M e with e along easy axis R R i and along other selected directions. The k derivative of the NL Bloch wave functions can be accurately evaluated by either a Here Vps and VR are the local part and the nonlocal part in 18 19 NL NL covariant finite difference formula or by the k·p method. separable form of the norm-conserving PS. FR and ER are For insulating systems both methods provide exactly the the separable nonlocal GIPAW projectors, accounting, re- same results; for metallic systems the covariant derivative is spectively, for the so-called paramagnetic and diamagnetic more involved and we apply just the k·p method. For the 13 contributions of the atomic site R. ferromagnetic Fe, Co, and Ni, calculations are carried out at H͑1͒ H͑1͒ ͑ ͒ Inserting + SO in Eq. 1 we obtain the experimental lattice constants. We consider 4s and 4d M = M + ⌬M + ⌬M + ⌬M , ͑9͒ states in the valence with nonlinear core correction. use a bare bare para dia relatively low cutoff of 90 Ry. In the case of Fe, the results do not change by more than 1% by including 3s and 3p in ␣ 1 ͳ ϫ ͓ H͑0͒ H͑0͔͒ʹ ͑ ͒ valence and working at 120 Ry. We use a Marzari-Vanderbilt Mbare = ͚ r r, + SO , 10 2 R i cold smearing of 0.01 Ry. We carefully test our calculations

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FIRST-PRINCIPLES THEORY OF ORBITAL MAGNETIZATION PHYSICAL REVIEW B 81, 060409͑R͒͑2010͒

TABLE I. Orbital magnetic moments M͑e͒·e in ␮B per atom of ferromagnetic metals parallel to the spin, denotes the experimental easy axis. The interstitial contribution is deﬁned ء .for different spin orientations e MT Ã by the difference between M͑e͒ and M =͚ ͐⍀ u ͑r͒rϫ͑−iٌ+k͒u ͑r͒dr, where ⍀ is a atom-centered orb nk s nk nk s sphere of radius RMT=2.0 rbohr. All theoretical values are based on the gradient corrected PBE functional. The decomposition of M according to Eq. ͑9͒ is shown Table II of the auxiliary material.

This method

Metal e Expt. 20 Total Interstitial MT FLAPW5

0.045 0.0433 0.0225 0.0658 0.081 ءbcc-Fe ͓001͔ bcc-Fe ͓111͔ 0.0660 0.0216 0.0444 0.073 0.0634 0.0122 0.0756 0.120 ءfcc-Co ͓111͔ fcc-Co ͓001͔ 0.0660 0.0064 0.0596 0.0868 0.0089 0.0957 0.133 ءhcp-Co ͓001͔ hcp-Co ͓100͔ 0.0867 0.0068 0.0799 0.050 0.0511 0.0008 0.0519 0.053 ءfcc-Ni ͓111͔ fcc-Ni ͓001͔ 0.0556 0.0047 0.0509 for k-point convergence. In all cases a 28ϫ28ϫ28 mesh sphere and thus leads to considerably improved ab initio Ϯ ␮ yields converged results within 0.0001 B. values. This result indicates the importance of the contribu- Table I reports our results for the orbital magnetization of tions from the interstitial regions when benchmarking and/or the three metals Fe, Co, and Ni, together with experimental developing improved DFT functionals for orbital magnetism. values and a recent calculation performed by full potential In the following we will show that the anisotropies in the linearized augmented plane wave ͑FLAPW͒.5 All theoretical orbital magnetizations are well described to allow us to cal- data were obtained using the PBE functional. LDA gives culate the electronic g tensor of paramagnetic systems, in Ϯ ␮ ͑ within 0.003 B the same values see Table II in the addi- order to understand the microscopic structure of radicals or tional material21͒. In order to evaluate the contribution of the paramagnetic defects in solids. From the orbital magnetiza- interstitial regions neglected so far in the literature ͑as in tion we can obtain the deviation of the g tensor, ⌬g␮␯ from ͒ ͑␣/ ͒͗ ͘ Refs. 5 and 6 , we have equally computed 2 L only the free-electron value ge =2.002 319 by the variation in M inside atomic spheres. Except for fcc-Co our results agree with a spin flip: very well with FLAPW calculations. For Ni the influence of contributions is indeed negligible, explaining the agreement 2 M͑e␯͒ − M͑− e␯͒ 2 ⌬g␮␯ =− e␮ · =− e␮ · M͑e␯͒, of early DFT calculations in this case. For the other ferro- ␣ S − ͑− S͒ ␣S magnets however it becomes evident from Table I that these ͑14͒ contributions can by no means be neglected. For Fe, e.g., the interstitial contribution is about 50% of that inside a MT where ␯, ␮ are Cartesian directions of the magnetic field, and the total spin S, respectively. To get the full tensor ⌬g␮␯, for ⌬ TABLE II. Principal values g in ppm for the diatomic mol- every paramagnetic systems we carry out three calculations ͑ ͒ ecules of the RnF-family calculated by LR Ref. 10 and with the by aligning the spin-quantization axis along the three Carte- ʈ ⌬ ͑⌬ ͒ current method. is symmetry axis of the dimer. g M gives the sian directions. contributions of ⌬M , ⌬M , and ⌬M to the g tensor. A bare para dia To evaluate the approach, we compute the g tensors of ͑small͒ relativistic mass correction term ⌬g ͑Ref. 10͒ is in- RMC selected diatomic radicals. An energy cutoff of 100 Ry is cluded in both sets of data. used in all molecular calculations. They are performed in a cubic repeated cell with a large volume of 8000 Å3 and the Linear response This method ⌬g͑⌬M͒ Brillouin zone is sampled only at the ⌫ point. For compari- 10 NeF ⌬gʈ −336 −328 −414 son, we also compute the g-tensor via the LR method, ⌬gЌ 52633 52778 2935 which we recently implemented in the QUANTUM-ESPRESSO package. For a wide range of molecular radicals including ArF ⌬gʈ −349 −343 −4450 almost all of the examples discussed in Ref. 10, the approach ⌬gЌ 42439 42519 2914 reproduces the values obtained via LR within a few ppm ͑see ⌬ KrF gʈ −360 −353 −968 also auxiliary Table I in Ref. 21͒. In Tables II and III we ⌬gЌ 59920 59674 −1918 report the calculated principal components of the computed g ⌬ XeF gʈ −358 −354 −3733 tensors for the RnF and PbF families. For the members of the ⌬gЌ 163369 158190 −55099 RnF family qualitative deviations are only observed if heavy

RnF ⌬gʈ −356 −299 −13670 elements like Xe and Rn are involved, showing in LR large ⌬ 5 ⌬gЌ 603082 488594 −255079 deviations gЌ of up to 10 ppm from ge for the corresponding ﬂuorides. The treatment of SO-coupling beyond

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⌬ TABLE III. Calculated principal values g in ppm for the di- leading within LR to diverging values gʈ. This failure of LR atomic molecules of the PbF-family. See Table II for details. is observed for all members of the PbF family, already for CF containing light elements exclusively. In contrast, our Linear response This method ⌬g͑⌬M͒ method circumvents perturbation theory, and predicts a nearly vanishing g value gʈ =g +⌬gʈ Ϸ0 along the bond di- ⌬ ϱ e CF gʈ − −1999719 −119746 rection of the diatomic molecules as expected analytically.23 ⌬gЌ 1920 −553 −240 In conclusion, we have shown how a recently developed

SiF ⌬gʈ −ϱ −1995202 −100021 formula for the orbital magnetization can be applied in an ab ⌬gЌ −480 −2470 −535 initio pseudopotential scheme whereby the spin-orbit coupling enters explicitly the self-consistent cycle. In compari- GeF ⌬gʈ −ϱ −1998078 −40609 son with linear response methods, our approach allows an ⌬gЌ −15505 −39101 −388 improved calculation of the electronic g tensor of paramag- ⌬ ϱ SnF gʈ − −1996561 −72464 netic systems containing heavy elements or with large devia- ⌬gЌ −64997 −142687 −5339 tions of the g tensor from the free-electron value. The latter PbF ⌬gʈ −ϱ −1999244 −90214 situation is encountered in many paramagnetic centers in sol- 24 ⌬gЌ −288383 −556326 −22476 ids, such as those exhibiting a Jahn-Teller distortion and/or containing transition metal impurities. In addition, our method provides improved orbital magnetizations with re- LR leads to considerably smaller values of ⌬gЌ, reduced by spect to the pre-existing approaches that neglect the contri- 3% ͑XeF͒ and 19% ͑RnF͒, respectively. Note that the recon- butions of the interstitial regions. This has been shown for struction terms, Eqs. ͑11͒–͑13͒, significantly contribute to the the highly ordered ferromagnets where the orbital contribu- g tensor. For the RnF family ͑see Table II͒ this is essential to tion is partially quenched by the crystal field. The presented obtain a value of ⌬gʈ Ϸ0 ͑Ref. 22͒ as also expected approach is perfectly suited to describe also the ferromag- analytically.23 netism of nanostructures where the orbital quench is weaker In contrast to the RnF family ͑five electrons in the p shell, and the orbital part of the magnetic moments becomes more 4 1 ͒ e a1 electronic configuration , the PbF family has only one dominant. electron within the p shell. Without SO-coupling the un- U.G. acknowledges financial support by the DFG ͑Grant paired electron occupies a degenerate e level. Consequently, No. GE 1260/3-1͒ and by the CNRS. D.C. acknowledges without SO, the highest occupied molecular orbital partial support from ENI. Calculations were performed at the ͑HOMO͒-lowest unoccupied molecular orbital ͑LUMO͒ gap IDRIS, Paris ͑Grant No. 061202͒ and at CINECA, Bologna between the unpaired electron and the empty levels is zero, ͑Grant No. Supercalcolo 589046187069͒.

1 13 ͉ ͘ R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 Given the set of GIPAW projectors ˜pR,n , the diamagnetic and ͑ ͒ NL ͚ ͉ ͘ ͗ ͉ 1993 ; D. Vanderbilt and R. D. King-Smith, ibid. 48, 4442 the paramagnetic term are ER = R,nm ˜pR,n eR,nm ˜pR,m and ͑1993͒. NL ͚ ͉ ͘␴ ͗ ͉ F = R,nm ˜pR,n ·fR,nm ˜pR,m . The expression of eR,nm and 2 ͑ ͒ R R. Resta, Rev. Mod. Phys. 66, 899 1994 . f , respectively, is given by Eqs. ͑10͒ and ͑11͒ of Ref. 10. 3 R,nm T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. 14 P. Giannozzi et al., J. Phys.: Condens. Matter 21, 395502 ͑2009͒ Rev. Lett. 95, 137205 ͑2005͒; D. Ceresoli, T. Thonhauser, D. http://www.quantum-espresso.org. Vanderbilt, and R. Resta, Phys. Rev. B 74, 024408 ͑2006͒. 15 N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 ͑1991͒. 4 D. Xiao, J. Shi, and Q. Niu, Phys. Rev. Lett. 95, 137204 ͑2005͒; 16 J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 ͑1981͒. J. Shi, G. Vignale, D. Xiao, and Q. Niu, ibid. 99, 197202 17 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, ͑2007͒. ͑ ͒ 5 First Principles Determination of Magnetic Anisotropy 1396 1997 . R. Wu, 18 and Magnetostriction in Transition Metal Alloys, Lecture Notes N. Sai, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B 66, 104108 ͑ ͒ in Physics 580 ͑Springer, Berlin, 2001͒. 2002 . 19 ͑ ͒ 6 S. Sharma, S. Pittalis, S. Kurth, S. Shallcross, J. K. Dewhurst, C. J. Pickard and M. C. Payne, Phys. Rev. B 62, 4383 2000 ; and E. K. U. Gross, Phys. Rev. B 76, 100401͑R͒͑2007͒. M. Iannuzzi and M. Parrinello, ibid. 64, 233104 ͑2001͒. 20 7 O. Eriksson, M. S. S. Brooks, and B. Johansson, Phys. Rev. B A. J. P. Meyer and G. Asch, J. Appl. Phys. 32, S330 ͑1961͒. 41, 7311 ͑1990͒; J. Trygg, B. Johansson, O. Eriksson, and J. M. 21 See supplementary material at http://link.aps.org/supplemental/ Wills, Phys. Rev. Lett. 75, 2871 ͑1995͒. 10.1103/PhysRevB.81.060409 for tables. 8 J.-M. Spaeth and H. Overhof, Point Defects in Semiconductors 22 Y. Y. Dmitriev et al., Phys. Lett. A 167, 280 ͑1992͒. and Insulators ͑Springer, Berlin, 2003͒. 23 Along the bond direction ͑in absence of SO͒ the angular momen- 9 G. Schreckenbach and T. Ziegler, J. Phys. Chem. A 101, 3388 tum is a good quantum number leading to gʈ Ϸge +2ml: for the ͑1997͒. PbF family gʈ Ϸ0 ͑the unpaired electron occupies the ml =−1 10 C. J. Pickard and F. Mauri, Phys. Rev. Lett. 88, 086403 ͑2002͒. p-like orbital͒; for the RnF family ml =0 and, thus, gʈ Ϸge. 11 S. Patchkovskii, R. T. Strong, C. J. Pickard, and S. Un, J. Chem. 24 The method is, e.g., applicable in case of some intrinsic defects Phys. 122, 214101 ͑2005͒. ͑silicon antisites͒ in the compound semiconductor SiC where LR 12 C. J. Pickard and F. Mauri, Phys. Rev. B 63, 245101 ͑2001͒; fails to achieve convergence with respect to k points; U. Gerst- Phys. Rev. Lett. 91, 196401 ͑2003͒. mann et al. ͑unpublished͒.

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