Issues with J-dependence in the LSDA+U method for non-collinear magnets
IIssues with J -dependence in the LSDA + U method for non-collinear magnets Eric Bousquet , and Nicola Spaldin Materials Department, University of California, Santa Barbara, CA 93106, USA Physique Théorique des Matériaux, Université de Liège, B-4000 Sart Tilman, Belgium and Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 10 CH-8093 Zurich, Switzerland
We re-examine the commonly used density functional theory plus Hubbard U (DFT + U ) methodfor the case of non-collinear magnets. While many studies neglect to explicitly include the exchangecorrection parameter J , or consider its exact value to be unimportant, here we show that in the caseof non-collinear magnetism calculations the J parameter can strongly affect the magnetic groundstate. We illustrate the strong J -dependence of magnetic canting and magnetocrystalline anisotropyby calculating trends in the magnetic lithium orthophosphate family LiMPO (M = Fe and Ni) anddifluorite family MF (M = Mn, Fe, Co and Ni). Our results can be readily understood by expandingthe usual DFT + U equations within the spinor scheme, in which the J parameter acts directly onthe off-diagonal components which determine the spin canting. Keywords: first-principles, LDA+U, non-collinear magnetism, magnetocrystalline anisotropy
Density functional theory (DFT) within the local den-sity (LDA) and generalized gradient (GGA) approxima-tions is widely used to describe a large variety of materi-als with good accuracy. The LDA and GGA functionalsoften fail, however, to correctly reproduce the propertiesof strongly correlated materials containing d and f elec-trons. The LDA + U approach – in which a Hubbard U re-pulsion term is added to the LDA functional for selectedorbitals – was introduced in response to this problem, andoften improves drastically over the LDA or GGA. Indeed,it provides a good description of the electronic propertiesof a range of exotic magnetic materials, such as the Mottinsulator KCuF and the metallic oxide LaNiO .Two main LDA + U schemes are in widespread usetoday: The Dudarev approach in which an isotropicscreened on-site Coulomb interaction U eff = U − J isadded, and the Liechtenstein approach in which the U and exchange ( J ) parameters are treated separately. TheDudarev approach is equivalent to the Liechtenstein ap-proach with J = 0 . Both the effect of the choice ofLDA+ U scheme on the orbital occupation and subse-quent properties , as well as the dependence of the mag-netic properties on the value of U , have recently beenanalyzed. There has been no previous systematic study,however, of the effect of the J parameter of the Liechten-stein approach in non-collinear magnetic materials. Herewe show that neither the approach of not explicitly con-sidering the J parameter (as in the Dudarev implemen-tation), nor the assumption that its importance is bor-derline – a common approximation is to use J (cid:39) U without careful testing – within the Liechtenstein imple-mentation are justified in the case of non-collinear mag-nets. We demonstrate that in the case of non-collinearantiferromagnets, the choice of J can strongly change theamplitude of the spin canting angle (LiNiPO ) or evenmodify the easy axis of the system (LiFePO and FeF ),with consequent drastic effects on the magnetic suscep-tibilities and magnetoelectric responses.First we remind the reader how the U and J parame-ters appear in the usual collinear spin LSDA + U formal- ism. The LSDA + U reformulation of the LSDA Hamilto-nian is usually written as: H LSDA + U = H LSDA + H U , (1)whith H σU = (cid:88) m ,m P m ,m V σm ,m , (2)where P is the projection operator, σ is the spin index,and (on a given atomic site): V ↑ ( ↓ ) m ,m = (cid:88) , (cid:0) V ee , , , − U δ , − V ee , , , + Jδ , (cid:1) n ↑ ( ↓ )3 , + (cid:0) V ee , , , − U δ , (cid:1) n ↓ ( ↑ )3 , + 12 ( U − J ) δ , (3)Here V ee , , , = (cid:10) m , m (cid:12)(cid:12) V eem ,m ,m ,m (cid:12)(cid:12) m , m (cid:11) are theelements of the screened Coulomb interaction (which canbe viewed as the sum of Hartree (direct) contributions V ee , , , and Fock (exchange) contributions V ee , , , and n σi,j are the d -orbital occupancies.In the case of non-collinear magnetism, the formal-ism is extended and the density is expressed in a two-component spinor formulation: ρ = (cid:18) ρ ↑↑ ρ ↑↓ ρ ↓↑ ρ ↓↓ (cid:19) = (cid:18) n + m z m x − im y m x + im y n − m z (cid:19) (4)where n is the charge density and m α the magnetiza-tion density along the α direction ( α = x, y, z ). Usingthe double-counting proposed by Bultmark et al. , theLSDA + U potential is then also expressed in the two-component spin space as: V i,j = (cid:32) V ↑↑ i,j V ↑↓ i,j V ↓↑ i,j V ↓↓ i,j (cid:33) (5)where V ↑↑ and V ↓↓ are equal to Eqs.3 and V ↑↓ ( ↓↑ ) m ,m = (cid:88) , (cid:0) − V ee , , , + Jδ , (cid:1) n ↑↓ ( ↓↑ )3 , (6) a r X i v : . [ c ond - m a t . s t r- e l ] N ov For collinear magnets, only V ↑↑ and V ↓↓ (Eqs. 3) arerelevant since n ↑↓ and n ↓↑ are equal to zero, and J affectsthe potential mainly through an effective U − J . However,in the case of non-collinear magnetism, the n ↑↓ and n ↓↑ and hence the V ↑↓ and V ↓↑ (Eqs. 6) are non-zero. Thenit is clear from Eqs. 6 that J acts explicitly on the off-diagonal potential components.Next, we show the effect of the choice of J parameterin the family of lithium orthophosphates, LiMPO (M= Ni and Fe) and in the family of difluorites MF (M= Mn, Co, Fe and Ni). The orthophosphates crystallizein the orthorhombic Pnma space group with C -type an-tiferromagnetic (AFM) order. The difluorites crystalizein the tetragonal P4 /mnm rutile structure with AFMorder. We performed calculations within the Liechten-stein approach of the DFT + U as implemented in theVASP code with U and J corrections applied to the3 d orbitals of the M cations. In all cases we relaxed theatomic positions until the residual forces on each atomwere lower than 10 µ eV/Å at the experimental volumeand cell shape reported in Tab. I, taking into accountthe spin-orbit interaction. We found good convergenceof the non-collinear spin ground state with a cutoff en-ergy of 500 eV on the plane wave expansion and a k-pointgrid of × × for the orthophophates and × × forthe difluorites. a b c Ref.LiFePO phosphates and MF difluorites. First, we focus on LiNiPO , which is known experi-mentally to be C -type AFM, with an easy-axis along the c direction and a small A-type AFM canting of the spinsalong the a direction ( C z A x ground state with mm’m magnetic point group) . Performing calculations withinthe LSDA + U method with J = 0 , we find that we cor-rectly reproduce the C z A x ground state with a rathersmall U sensitivity of the magnetocrystalline anisotropyenergy (MCAE) and the spin canting; this finding isconsistent with a previous report using the GGA func-tional . However, our calculated canting angle of 1.6 ◦ for U = 5 eV and J = 0 eV severely underestimatesthe experimental value of 7.8 ◦ . In Fig.1 (a) we showthe evolution of the canting angle with J at U = 5 eV.We find that the canting angle is extremely sensitive tothe value of J – in fact it is ∝ J – changing from 1.6 ◦ at J = 0 eV to 7.8 ◦ at J = 1 . eV. To reproduce theexperimental value of the canting angle we need to usethe rather large J value of 1.7 eV. The dependence ofthe canting angle on J is consistent with Eqs. 6, as the FIG. 1. (a) Calculated LSDA + U canting angle of LiNiPO versus J for U = 5 eV. The experimental value of the cantingangle is equal to 7.8 ◦ . (b) Energy versus canting angle inLiNiPO for U = 5 eV and J = 0 eV (red circles), U = 5 eVand J = 1 eV (blue triangles), U eff = 4 eV (green crosses) and U = 5 eV and J = 1 eV but by fixing J = a and b orientations of the magnetic moments ofLiFePO . The experimental b orientations is taken as energyreference. off-diagonal elements n ↑↓ and n ↓↑ are non-zero when thespins cant away from the easy axis.In Fig. 1 (b) we report the energy versus the cantingangle in LiNiPO for U = 5 eV and different values of J .We see that as J is increased from J = 0 eV to J = 1 eV(red circles and blue triangles) the minimum of the en-ergy shifts to larger canting angle, with a stronger gainof energy with respect to the uncanted reference. Whenperforming the same calculation with U eff = U = 5 eV and J = 0 eV, which is formallyequivalent to the Dudarev approach with U eff = 5 eV.These comparisons confirm that varying U has a minimaleffect on the canting angle in LiNiPO and also that theuse of the Liechtenstein treatment of J is extremly impor-tant. To further confirm the direct relationship betweenthe spin canting and the J parameter, we performed thesame calculations with U = 5 eV and J = 1 eV but weartificially fixed J = 0 eV only in Eqs. 6 (pink squares inFig.1 (b)). We clearly see that the energy versus cantingangle is strongly affected by this modification and in factthe canting is almost removed.Similar J dependence of the canting angle was also re-ported previously for Ni in BaNiF ; in Ref. 19 it wasfound that at U = 5 eV, the canting varies from 2 ◦ to 3 ◦ when J is varied from 0 eV to 1 eV. In both LiNiPO andBaNiF the Ni ion is divalent, with a d configuration,and octahedrally coordinated. To investigate the gener-ality of this behavior, we next consider the case of thecanted-spin antiferromagnet NiF , in which the Ni ion isin the same coordination environnement as in BaNiF .Experimentally, NiF has the spins aligned preferentiallyin the plane perpendicular to the c axis with a slight cant-ing from antiparallel alignment by an estimated ∼ ◦ atlow temperatures . Performing LSDA + U calculationsat the experimental volume and with U = 5 eV and J = 0 eV we indeed obtain the easy axis perpendicular to the c axis and a small canting of 0.3 ◦ , in excellent agreementwith the experiments. In contrast to the case of LiNiPO ,however, we find that the amplitude of the canting angleis almost insensitive to the value of J with just a smalltendency to be reduced when J increased. This insen-sitivity of the canting angle to the value of J in NiF can be understood from the fact that in this compoundthe magnetism is almost collinear, and therefore the off-diagonal elements of the occupation matrix, n ↑↓ and n ↓↑ ,are close to zero. Inspection of Eqs. 3 then shows thatthe effect of J is reduced largely to the diagonal part ofthe potential where the U parameter is dominant.To summarize our findings for the Ni-based com-pounds, in cases where the experimental canting is large(2-3 ◦ ) we find a strong J -dependence of the canting an-gle, which increases with increasing J ; when the cantingis weak experimentally the J -dependence is much weaker. FIG. 2. Magnetocrystaline anisotropy energy versus the J parameter of (a) FeF (Experimental value from Ref.20), (b) NiF ,(c) MnF (Experimental value from Ref.21) and (d) CoF (“sc“ are calculations with Co semi-cores while ”no sc” are calculationswithout Co semi-cores). The MCAE reported here is the energy between the a and c orientation of the spins, the energy of the c orientation is taken as reference. Next we analyse the effect of J on the behavior onthe corresponding divalent iron compounds. We beginwith LiFePO , which is known experimentally to be a C -type AFM with an easy axis along the b direction and noobserved canting of the spins ( C y ground state with mmm’ magnetic point group). Our calculations withinthe LSDA + U functional at the commonly used values of U = 4 eV and J = 0 eV for Fe yield the correct C -type AFM order but find the easy axis incorrectly alongthe a direction. Now we switch to J (cid:54) = 0 eV and reportin Fig. 1.c the MCAE between the b and a directions,calculated by turning all the spins homogenously fromthe C y to the C x direction. We find that the MCAE isapproximately linear with J , but with rather dramaticqualitative dependence: while at J = 0 eV the easy axisis along the a direction (negative MCAE) the MCAE isalmost reduced to zero around J = 0 . eV and the easyaxis changes to the b direction for J > ∼ . eV (positiveMAE). To reproduce the experimental easy axis ( C y ) avalue of J greater than 0.58 eV is required. In the cases where the correct easy axis is reproduced ( C y ) we do notobserve any canting of the spins, in agreement with theexperimental magnetic point group mmm’ .As a second example with Fe , we analyse the effect of J on the MCAE of FeF . Experimentally FeF is knownto have its spin magnetization parallel to the tetragonal c axis with a rather large MCAE of about +4800 µ eV .In Fig. 2.a we report the LSDA + U MCAE energies withrespect to J at four different values of U (3, 4, 5 and 6eV). All the calculations with J = 0 eV give the wrongeasy axis (spins are perpendicular to c ) with a huge errorin the MCA energy (MCAE from -16000 to -26000 µ eVfor U going from 3 to 6 eV). Increasing the value of J inthe range of 0–0.5 eV has the tendency to strongly reducethis error with a linear increase of the MCAE with J aswe found above for LiFePO . However beyond J (cid:39) . the increase of the MCAE is reduced and the evolutionbecomes more complex with the appearance of two max-ima before a drastic decrease beyond J (cid:39) . eV. Thecorrect easy axis (MCAE > ) is only obtained for a verysmall range of U and J values, and the amplitude of theMCAE is correct over an even smaller range. This J de-pendence of the MCAE is again consistent with Eqs.3-6.From Eq.4 it is clear that when changing the orienta-tion of the spins from the z axis to the x or y axis theoff-diagonal parts of Eq.4 become non-zero resulting in adirect effect of J on the MCAE from Eqs.6.We also performed the same analysis of the MCAEfor NiF (Fig.2.b), MnF (Fig.2.c) and CoF (Fig.2.d).MnF and CoF have the same easy axis as FeF whileNiF has its easy axis perpendicular to the c direction.The easy axis is well reproduced for all three compoundsat J = 0 eV. As for FeF , the amplitudes of the MCAEdepend strongly on J but with a completely differenttrend in each compound. For MnF and FeF the experi-mental value can be reproduced by adjusting the values of U and J . In the case of CoF and NiF no experimentalvalues are available. For CoF we also performed calcula-tions with and without Co semi-cores states (Fig.2.d) andfind a strong difference in the magnitude of the MCAE forthe two cases. For FeF we also performed calculationswithin the GGA functional (black pentagons in Fig.2.a)and obtained a completely different J dependence thanthose calculated with the LDA functional. These com-parisons illustrate the difficulty of extracting a generalrule about the J dependence of the MCAE.Our results reveal a problem with the predictability ofthe LSDA + U method for non-collinear magnetic materi- als: A strong dependence of the MCAE and spin cantingangles on the values of U and particularly J that areused in the calculation. Since properties such as magne-tostriction, piezomagnetic response, magnetoelectric re-sponse and exchange bias coupling are directly related toMCAEs and spin canting, it is of primary importanceto reproduce these quantities accurately. At the mo-ment, the most reliable, although not entirely satisfac-tory, option appears to be a fine tuning of the U and J parameters by adjustment to reproduce experimentallymeasured anisotropies and canting angles; there is someevidence to suggest that properties such as magnetoelec-tric responses are then in turn well reproduced . Futurestudies might explore methodologies for self-consistentcalculation of the J parameter, or the predictions of newdescriptions of the exchange and correlation such as thehybrid functionals . On the flip side, it is clear that non-collinear magnetic systems provide a challenging case fortesting the correctness of new exchange correlation func-tionals within the density functional formalism. Acknowledgments:
This work was supported by theDepartment of Energy SciDAC DE-FC02-06ER25794.We made use of computing facilities of TeraGrid at theNational Cen-ter for Supercomputer Applications and ofthe California Nanosystems Institute with facilities pro-vided by NSF grant No. CHE-0321368 and Hewlett-Packard. EB also acknowledges FRS-FNRS Belgium andthe ULg SEGI supercomputer facilities. A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.Rev. B , R5467 (Aug 1995). K.-W. Lee and W. E. Pickett, Phys. Rev. B , 165109(Oct 2004). S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.Humphreys, and A. P. Sutton, Phys. Rev. B , 1505 (Jan1998). P. Baettig, C. Ederer, and N. A. Spaldin, Phys. Rev. B ,214105 (2005). E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, Phys.Rev. B , 035103 (Jan 2009). S. Y. Savrasov, A. Toropova, Kat, K. M. I., L. A. I., A. V.,and G. Kotliar, Z. Kristallogr. , 473 (2005). F. Bultmark, F. Cricchio, O. Grånäs, and L. Nordström,Phys. Rev. B , 035121 (2009). G. Kresse and J. Furthmüller, Phys. Rev. B , 11169 (Oct1996). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (Jan1999). We note that LSDA + U double-couting term taking intoaccound the magnetization density as proposed by Bult-mark et al. is mandatory within non-collinear magnetismcalculations. This is not necessarily done in the presentimplementation of other codes. V. A. Streltsov, E. L. Belokoneva, V. G. Tsirelson, andN. K. Hansen, Acta Cryst. B , 147 (Apr 1993). I. Abrahams and K. S. Easson, Acta Cryst. C , 925(1993). M. T. Hutchings, M. F. Thorpe, R. J. Birgeneau, P. A. Fleury, and H. J. Guggenheim, Phys. Rev. B , 1362 (Sep1970). M. J. M. de Almeida, M. M. R. Costa, and J. A. Paixão,Acta Cryst. B , 549 (1989). T. Oguchi, Phys. Rev. , 1063 (Aug 1958). N. J. O’Toole and V. A. Streltsov, Acta Cryst. B , 128(2001). T. B. S. Jensen, N. B. Christensen, M. Kenzelmann, H. M.Rønnow, C. Niedermayer, N. H. Andersen, K. Lefmann,J. Schefer, M. v. Zimmermann, J. Li, J. L. Zarestky, andD. Vaknin, Phys. Rev. B , 092412 (2009). K. Yamauchi and S. Picozzi, Phys. Rev. B , 024110(2010). C. Ederer and N. A. Spaldin, Phys. Rev. B , 020401 (Jul2006). M. E. Lines, Phys. Rev. , 543 (Apr 1967). U. Gäfvert, L. Lundgren, P. Nordblad, B. Westerstrandh,and . Beckman, Sol. State Comm. , 9 (1977). Zimmermann, A. S., Van Aken, B. B., Schmid, H., Rivera,J.-P., Li, J., Vaknin, D., and Fiebig, M., Eur. Phys. J. B , 355 (2009). G. Liang, K. Park, J. Li, R. E. Benson, D. Vaknin, J. T.Markert, and M. C. Croft, Phys. Rev. B , 064414 (2008). C. Rudowicz, J. Phys. Chem. Solids , 1243 (1977). R. C. Ohlmann and M. Tinkham, Phys. Rev. , 425(1961). K. Delaney, E. Bousquet, and N. A. Spaldin,arXiv:0912.1335v2(2010). J. Heyd and G. E. Scuseria, The Journal of Chemical
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