Origin of the spin reorientation transitions in (Fe$_{1-x}$Co$_{x}$)$_{2}$B alloys

0.8 MB·English

by  Kirill D. Belashchenko

Origin of the spin reorientation transitions in (Fe Co ) B alloysa)
1 x x 2
−
Kirill D. Belashchenko,1 Liqin Ke,2 Markus D¨ane,3 Lorin X. Benedict,3 Tej Nath Lamichhane,2,4 Valentin
Taufour,2,4 Anton Jesche,2,4 Sergey L. Bud’ko,2,4 Paul C. Canfield,2,4 and Vladimir P. Antropov2
1)Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,
University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
2)Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011, USA
3)Lawrence Livermore National Laboratory, Livermore, California 94550, USA
4)Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA
5
(Dated: 4 February 2015)
1
0 Low-temperature measurements of the magnetocrystalline anisotropy energy K in (Fe Co ) B alloys are
1−x x 2
2
reported, and the origin of this anisotropy is elucidated using a first-principles electronic structure analysis.

b The calculated concentration dependence K(x) with a maximum near x=0.3 and a minimum near x=0.8
e isinexcellentagreementwithexperiment. Thisdependence istraceddowntospin-orbitalselectionrulesand
F
the filling of electronic bands with increasing electronic concentration. At the optimal Co concentration, K

3 depends strongly on the tetragonality and doubles under a modest 3% increase of the c/a ratio, suggesting
that the magnetocrystalline anisotropy can be further enhanced using epitaxial or chemical strain.

]
i
c
s Magnetocrystalline anisotropy (MCA) of a magnetic latedusingRef.10. Field-dependentmagnetizationmea-
-
l materialisone ofits keypropertiesforpracticalapplica- surements were performed in a Quantum Design MPMS
r
t tions, large easy-axis anisotropy being favorable for per- at 2 K in fields up to 5.5 T. The MCA energy K was
m manent magnets.1 Intelligent search for new materials determined as the area between the two magnetization
. requires understanding of the underlying mechanisms of curves,with the field paralleland perpendicular to the c
t
a MCA.This canbe particularlyfruitful forsubstitutional axis,takenatthe same temperature.6 The resultsshown
m alloyswhosepropertiescanbe tunedbyvaryingthecon- in Fig. 1 (blue stars) measured at 2 K confirm the non-
- centrations of their components. The analysis is often monotonicconcentrationdependence,ingoodagreement
d
relatively simple in insulators, where MCA is dominated with the measurements at 77 K from Ref. 2.
n
by single-ion terms which can be deduced from crystal-
o
field splittings and spin-orbital (SO) selection rules. In
c
[ contrast, in typical metallic alloys the band width sets

the largest energy scale, and MCA depends on the de-
2
tails of the electronic structure.
v
The (Fe Co ) B solid solution2–6 (space group
3 1−x x 2
8 I4/mcm7) is a remarkable case in point. Fe2B has a
4 fairlystrongeasy-planeMCA,andCo B,atlowtemper-
2
3 atures, a small easy-axis MCA. However, the alloy has a
0 substantial easy-axis MCA around x = 0.3,2 making it
1. a potentially useful rare-earth-free8 permanent magnet.
0 At x 0.5 the MCA again turns easy-plane, peaks at
5 x = 0.≈8, and then turns easy-axis close to x = 1. These
1
three spin reorientation transitions must be related to
:
v the continuous evolution of the electronic structure with
i concentration. The goalof this Letter is to elucidate the FIG. 1. Calculated (black circles) and experimental (blue
X
origin of this rare phenomenon. stars) MCA energy K in (Fe1−xCox)2B alloys. Gray curve:
r First, we report the results of experimental mea- KSO. The other lines show thespin decomposition Kσσ′.
a
surements at low temperatures. Single crystals of
(Fe Co ) B were grown from solution growth out of Density-functional calculations using several different
1−x x 2
anexcessof(Fe,Co)whichwasdecantedinacentrifuge.9 methodsshowthatthechoiceoftheexchange-correlation
The single crystals were grown as tetragonal rods which potentialandother computationaldetails stronglyaffect
werecutusingawiresawtogivethemtheshapeofarect- the calculated MCA in Fe B and Co B. We have ascer-
2 2
angular prism. The demagnetization factor was calcu- tainedthatthis sensitivity is largelydue to the variation
of the exchange splitting, which controls the position of
the Fermi level relative to the minority-spin bands. The
systematic variation of MCA with x is also controlled
a)This article has been accepted by Applied Physics by the Fermi level shift. This continuous variation is,
Letters. After it is published, it will be found at therefore,reliablypredictedaslongastheendpointsare
http://scitation.aip.org/content/aip/journal/apl. correctly fixed using experimental input.
2
The results reported below were obtained using the pure Fe B and Co B agree very well with Hamiltonian
2 2
Green’s function-based formulation of the tight-binding LMTO results. The concentration dependence of K is
linear muffin-tin orbital (GF-LMTO) method.11 Substi- shown in Fig. 1. The agreement with low-temperature
tutional disorder was treated using our implementation experimental data is remarkably good, suggesting that
of the coherent potential approximation (CPA),12 with the electronic mechanisms of MCA are correctly cap-
the SO coupling included perturbatively as described in tured in the calculations. If the spin moment of Co is
the supplementary material.13 not corrected by scaling the exchange-correlation field,
The lattice constants and the internal coordinates are the downward trend in K at the Co-rich end continues
linearly interpolated between the experimental data for tolargenegativevaluesindisagreementwithexperiment.
the end compounds extrapolated to zero temperature:2 In 3d systems the SO band shifts are usually well
a = 5.109 and 4.997 ˚A, c = 4.249 and 4.213 ˚A, and described by second-order perturbation theory, except
u = 0.166 and 0.168 for Fe2B and Co2B, respectively. perhaps in small regions of the Brillouin zone. Conse-
Theexchangeandcorrelationaretreatedwithinthegen- quently, when MCA appears in second order in SO cou-
eralized gradient approximation (GGA).14 pling (as in the tetragonal system under consideration),
The correct exchange splitting at the end points can the anisotropyof the expectation value of the SO opera-
be enforced by using the experimental data2–6 for the tor ∆E = V V is approximately equal to
SO SO x SO z
magnetization M: 1.9 µB/Fe in Fe2B and 0.76 µB/Co 2K.13 (Here xh or zi s−hohws thie orientation of the magne-
in Co2B. In Fe2B it is only slightly overestimated, but tizationaxis.) We thereforedenote KSO =∆ESO/2and
in Co2B it is much too large at about 1.1 µB/Co in all use the expression2 SL = Lz′ ↑↑ Lz′ ↓↓+ L+ ↓↑+
density-functionalcalculations. Therelativelysmallspin L toseparatethhecointrihbutiions−tohK biypaihrsofispin
− ↑↓
moment of Co indicates a pronounced itinerant charac- chhaninels. Here we use Lz′ to denote the component of
ter ofmagnetisminCo2B,whichtends to be sensitive to L parallel to the magnetization axis, to avoid confusion
quantum spin fluctuations. Therefore, we introduced a with the crystallographic z axis; L are the usual linear
±
scaling factor for the local part of the effective magnetic combinationsoftheothertwo(primed)componentsofL.
field in the GGA functional for the Co atoms and ad- The contributions Kσσ′ to KSO are accumulated as en-
justed it to match the experimental magnetization. The ergyintegralstakingintoaccounttheenergydependence
resultingscalingfactorof0.80wasthenusedforCoatoms ofthe SO couplingparameters. The resultsforK and
SO
at all concentrations. The spin moments on different Kσσ′ areshowninFig.1. First,weseethatKSO isclose
atoms and the total spin magnetization obtained in this toK,confirmingthevalidityofthisanalysis. Second,the
way are shown in Fig. 2. While the Fe spin moment is nonmonotonic concentration dependence of K is almost
almost constant, the Co spin moment declines with x. entirely due to K for x < 0.7. While K is sizeable,
↓↓ ↑↑
Moreover,this decline accelerates at x>0.6, which is in itdepends weaklyonxint∼his region. Additionaldiscus-
excellent agreement with experimental∼data.3 sion about the spin decomposition of MCA is provided
in the supplementary material.13
Letusnowanalyzetheelectronicstructureandthede-
tailsofSOcoupling-inducedmixingfortheminority-spin
electrons. Toresolvetheminority-spincontributiontoK
by wave vector k, we calculated the minority-spin spec-
tralfunctioninthepresenceofSOcouplingandfoundits
first energy moment at each k. Fig. 3 shows the differ-
ence ofthese integralsformagnetizationalongthe xand
z axes at three key concentrations: pure Fe B (x = 0),
2
themaximumofK (x=0.3)anditsminimum(x=0.8).
We have checked that the Brillouin zone integral of the
k-resolved MCA energy (summed up over both spins)
agrees almost exactly with the value of K calculated in
the usual way.
Fig.3 showsthat the MCAenergy accumulatesovera
FIG. 2. Spin moments on different atoms and the total spin fairly large part of the Brillouin zone. At x=0 negative
magnetization M per transition-metal atom. contributions to K dominate over most of the Brillouin
zone. At x = 0.3 both positive and negative contribu-
The MCA energy K was obtained by calculating the tions are small. At x = 0.8 there are regions with large
single-particle energy difference for in-plane and out-of- positive and large negative contributions. Overall,it ap-
planeorientationsofthemagnetizationwhilekeepingthe pears that the most important contributions come from
LMTO chargesfixedat their self-consistentvalues found the vicinity of the ΓXM (kz = 0) plane and from the
without SO coupling. A uniform mesh of 303 points in vicinity of the ΓH (kx =ky =0) line.
thefullBrillouinzoneprovidedsufficientaccuracyforthe The partial minority-spin spectral functions for the
k integration. We have verified that the values of K for transition-metal site are displayed in Fig. 4 (panels (a)-
3
FIG. 3. Brillouin zone map of the k-resolved minority-spin contribution to K at: (a) x=0, (b) x=0.3, (c) x=0.8. Half of
theBrillouin zone is shown; thetop face of theplot is kz =0. Points H (same as M) and X are shown in panel(b). The color
intensity indicates the magnitudeof negative (blue) and positive (red) values.
(c)) along the important high-symmetry directions for contributiontoK issmall(Fig.3b). Thesetwocasesare
the same three concentrations. The coloring in this fig- sketched in Fig. 4d.
ure resolves the contributions from different 3d orbitals.
Atstilllargerxtheoddband5getsgraduallyfilled,ac-
At x = 0 the spectral function resolves the conventional
tivating the negative contribution to K from the mixing
electronicbandsofpureFe B(animaginarypartof0.004
2 ofband5withemptyevenbands6-7. Thistrendcontin-
eV is added to the energy to acquire them). At x = 0.3
ues till about x = 0.8, where an even pair of bands 6-7
and x = 0.8 substitutional disorder broadens the bands
(degenerateatΓ)beginsto fill(Fig.4c). Mixingofthese
by a few tenths of an electronvolt, but their identity is bands by Lˆ leads to an intense positive contribution to
z
in most cases preserved. Thus, we will discuss the SO-
K neartheΓpoint(clearlyseeninFig.3c),andthetrend
induced band mixing in the alloy, bearing in mind that
reverses again. Thus, the nonmonotonic dependence of
band broadening should reduce the values of MCA at
K in the whole concentration range is explained. Note
intermediate x.
that if the exchange-correlation field on the Co atoms
Aswehavelearnedabove,thedominantconcentration is not scaled down to bring the magnetization in agree-
dependence of K comes from the Lz′ ↓↓ term, where z′ ment with experiment, the exchange splitting remains
h i
is the magnetization axis. The electronic states on the toolarge,andbands6-7remainemptyuptox=1. Asa
whole ΓXM plane can be classified as even or odd under result,withoutthiscorrectiontheuninterruptednegative
reflection z z. Even (odd) states have m = 0, 2 trend brings K to large negative values in disagreement
→ − ±
(m = 1) character and appear red and green (blue) in with experiment.
±
Fig. 4. States of different parity do not intermix on this
ToassesstheeffectofatomicrelaxationsonK,weop-
plane in the absence of SO coupling, as is clearly seen
timizedallinequivalentstructureswithtwoformulaunits
in Fig. 4a. The selection rules for SO coupling of the per unit cell using the VASP code15 and the GGA. The
minority-spinstatesfollowfromthedefiniteparityofthe
volumes were fixed at the same values as in the CPA
components of Lˆ under reflection. Lˆ (even; relevant for
z studies at the same x, while the cell shape and inter-
M z)onlymixesstates ofthe sameparityonthe ΓXM
nal coordinates were relaxed. Since all these supercells
k
plane,ormoregenerallyorbitalsofthesamem. Coupling
preserve the σ reflection plane, the displacements of Fe
z
betweenstates ofthe m= 2 character(red)is stronger
andCo atomsareconfinedto the xy plane. Alldisplace-
±
compared to states of the m = 1 character (blue). In ments were less than 0.025 ˚A. One of the two structures
±
contrast, Lˆx (odd; relevant for M k x) couples states of at x = 0.5 breaks the C4 symmetry. For this structure
theoppositeparityonthe ΓXMplane,ormoregenerally wetook the averageofthe energiesforM xandM y
orbitals m and m 1. All these couplings contribute to as the in-plane value in the calculation ofkK, which ckor-
±
K only when the Fermi level lies between the two states responds to the averaging over different orientations of
thatarebeingcoupled. WheneverLˆz orLˆx couplessuch the same local ordering.
states, there is a negative contribution to the energy of
The changes in the absolute value of K due to the re-
the system with the corresponding direction of M.
laxation and its values (meV/f.u.) in the relaxed struc-
With the help ofFig.4 we cannowdeduce whichcou- tures were found to be: 26% and 0.11 in Fe B, 6%
2
plings contribute to K at different concentrations. At and 0.25 in Fe Co B,− 13% and−0.15 in the Fe−CoB
1.5 0.5
x = 0 the Fermi level lies between the filled even states [100] superlattice, 6% and−0.12 in the FeCoB [110] su-
(bands 1-2) and empty odd states (bands 3-4) near Γ. perlattice, 11% and 0.31 in Fe Co B), and 19%
0.5 1.5
Coupling of these states by Lˆ contributes to negative and 0.04−in Co B. W−hile MCA tends to be larg−er for
x 2
−
K. Filling of the hole pocket at Γ (bands 3-4) with in- ordered structures, the concentration trend in supercell
creasing x suppresses this contribution. At x = 0.3 the calculations agrees well with the CPA results for disor-
odd bands 3-4 are filled (Fig. 4b), and the minority-spin deredalloys. Althoughthissetofunitcellsislimited,the
4
FIG. 4. (a-c) Minority-spin partial spectral functions for the transition-metal site in the absence of SO coupling at (a) x=0,
(b) x = 0.3, and (c) x = 0.8. Energy is in eV. (d) Level diagram and SO selection rules at the Γ point (bands 1-4). Color
encodes the orbital character of the states. The intensities of the red, blue and green color channels are proportional to the
sum of xy and x2−y2 (m=±2),sum of xz and yz (m=±1), and z2 (m=0) character, respectively.
resultssuggestthatlocalrelaxationsdonotqualitatively minority-spinelectronic bands of particularorbitalchar-
change the concentration dependence shown in Fig. 1. acter. Near the optimal 30% concentration of Co, the
Larger positive values of K are favorable for perma- MCAenergyispredictedtoincreasequicklywiththec/a
nentmagnetapplications. Ourelectronicstructureanal- ratio, which could be implemented by epitaxial strain or
ysis shows that the maximum near x 0.3 corresponds a suitable chemical doping.
≈
to the optimal band filling. Further raising of K re- The work at UNL was supported by the National Sci-
quires favorable changes in the band structure, which ence Foundation through Grant No. DMR-1308751 and
could be induced by epitaxial or chemical strain. We performedutilizingtheHollandComputingCenterofthe
therefore considered the dependence of K at x = 0.3 on University of Nebraska. Work at Ames Lab and LLNL
the volume-conservingtetragonaldistortion. The results was supported in part by the Critical Materials Insti-
plotted in Fig. 5 show a very strong effect: K is doubled tute, an Energy Innovation Hub funded by the US DOE
under a modest 3% increase in c/a due to the sharply and by the Office of Basic Energy Science, Division of
increasing spin-flip contributions. A more detailed anal- Materials Science and Engineering. Ames Laboratory is
ysisshowsthatthe latterislargelyduetotheincreasein operated for the US DOE by Iowa State University un-
the c parameter. On the other hand, the minority-spin der Contract No. DE-AC02-07CH11358. Lawrence Liv-
contributionincreaseswithdecreasingvolume. Thus,in- ermoreNationalLaboratoryisoperatedforthe USDOE
creasingcanddecreasingabothhaveapositiveeffecton under Contract DE-AC52-07NA27344.
MCA. This enhancementcould be achievedthroughepi-
taxial multilayer engineering. The search for a suitable
alloyingelementtoenhancethec/aratioisaninteresting 1J.StohrandH.Siegmann,inMagnetism: Fromfundamentalsto
subject for further investigation. Nanoscale Dynamics (Springer,Berlin,2006),p.805.
2A.Iga,Jpn.J.Appl.Phys.9,415(1970).
3M. C. Cadeville and I. Vincze, J. Phys.: Metal Phys. 5, 790
(1975).
4L. Takacs, M. C. Cadeville, and I. Vincze, J. Phys. F: Metal
Phys.5,800(1975).
5W. Coene, F. Hakkens, R. Coehoorn, D. B. de Mooij, C. de
Waard,J.FidlerandR.Gr¨ossinger,J.Magn.Magn.Mater.96,
189(1991).
6M. D. Kuz’min, K. P. Skokov, H. Jian, I. Radulov and O. Gut-
fleisch,J.Phys.: Condens.Matter26,064205(2014).
7C. Kapfenberger, B. Albert, R. P¨ottgen, and H. Huppertz, Z.
Kristallogr.221,477(2006).
8L.H.Lewis andF. Jim´enez-Villacorta,Metall. Mater.Trans.A
44,2(2013).
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10A.Aharoni,J.Appl.Phys.83,3432(1998).
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Electronicstructureofdisordered alloys, surfacesandinterfaces
(Kluwer,Boston,1997).
FIG. 5. MCA energy K as a function of the c/a ratio at 12L.Ke,K.D.Belashchenko, M.vanSchilfgaarde,T.Kotani,and
x = 0.3 (the value c/a = 1 is assigned to the unstrained V.P.Antropov,Phys.Rev.B88,024404 (2013).
lattice). KSO and Kσσ′ are also shown. 13Seetheattachedsupplementarymaterialforimplementationde-
tails,explanation about KSO,andthe discussionoforbital mo-
mentanisotropy.
In conclusion, we have explained how the spin re- 14J.P.Perdew,K.Burke,andM.Ernzerhof,Phys.Rev.Lett.77,
orientation transitions in (Fe1−xCox)2B alloys originate 3865(1996); 78,1396(1997).
in the SO selection rules and the consecutive filling of 15G.KresseandJ.Furthmu¨ller,Comput.Mat.Sci.6,15(1996).
1
Supplemental Material:
Origin of the spin reorientation transitions in (Fe Co ) B alloys
1 x x 2
−
I. IMPLEMENTATION OF SPIN-ORBIT COUPLING IN GREEN’S FUNCTION-BASED LINEAR MUFFIN-TIN
ORBITAL (LMTO) METHOD
Magnetocrystalline anisotropy (MCA) of substitutional alloys is often studied using the fully-relativistic Korringa-
Kohn-RostockerorGreen’sfunction-basedLMTO(GF-LMTO)methodcombinedwiththecoherentpotentialapprox-
imation (CPA).1–4 Here we employed a perturbative implementation of spin-orbit coupling in GF-LMTO. Dealing
with conventional spinor wave functions, it simplifies the analysis of MCA without a significant loss of accuracy
compared to the fully-relativistic approach.
A. Relation between the Hamiltonian and Green’s function-based LMTO formulation
The so-called orthogonal Hamiltonian and Green’s function-based LMTO formulations are equivalent to second
order in (E E ) where E is the linearization energy.5,6 The Hamiltonian in the orthogonal basis is
ν ν
−
H =C+√∆S(1 γS)−1√∆, (1)
orth
−
where C, ∆, and γ are the diagonal matrices containing the LMTO potential parameters, and S is the structure
constant matrix. On the other hand, in GF-LMTO the Green’s function
G(z)=λ(z)+µ(z)[P(z) S]−1µ(z), (2)
−
is constructed using the diagonal matrices containing the potential functions
z C √∆ γ
P(z)= − , µ(z)= , λ(z)= . (3)
∆ +γ (z C) ∆+γ(z C) ∆+γ(z C)
l l
− − −
These definitions are related through energy derivatives (denoted by an overdot):
1
µ2 =P˙, λ= P¨/P˙. (4)
−2
TheserelationsguaranteethatG(z)doesnothavepolesatthepointswhereP(z)hassimplepoles. Itisstraightforward
to show that G(z)=(z H )−1, which means that G(z) in (2) is the exact resolvent of H .
orth orth
Third-orderparametri−zationofthepotentialfunctions5isoftenconsiderablymoreaccuratecomparedtothesecond-
order one. The accuracy of this parametrization is similar to that of the three-center LMTO Hamiltonian, although
they are not exactly equivalent. In the nearly orthogonalLMTO basis, the Hamiltonian and overlap matrices are7
H =H +(H E )E p(H E ),
orth orth ν ν orth ν
− −
O =1+(H E )p(H E ) (5)
orth ν orth ν
− −
where E and p are diagonalmatrices containing the linearization energies and the LMTO parameters p= φ˙2 , and
ν
h i
H is the same as in (1). The generalized eigenvalue problem is then written as
orth
det[(H E) (H E )(E E )p(H E )]=0 (6)
orth orth ν ν orth ν
− − − − −
The third-order term can be treated as a perturbation, and the variational correction formally amounts to the
substitution Eˆ1 Eˆ1+(E E )3p in the second-order eigenvalue equation det(H E)=0.
ν orth
In the GF-LM→TO formula−tion, third-order accuracy is similarly achieved by redefinin−g the potential functions:5,7
1z˜′′
P˜ =P(z˜), µ˜ =√z˜′µ(z˜), λ˜ = +z˜′λ(z˜) (7)
−2 z˜′
wherez˜=z+p(z E )3andz˜′ =dz˜/dz. Thesedefinitionsaredesignedtopreservetherelations(4). Ifthethird-order
ν
−
Green’s function G˜ is defined similarly to (2) but with the third-order potential functions (7), it can be shown that
1z˜′′
G˜ = +√z˜′(z˜ H )−1√z˜′. (8)
−2 z˜′ − orth
2
A solutionof det(H z˜)=0 correspondsto a pole of G˜, and the factors√z˜′ guaranteethat the residue at that
orth
−
pole is equal to 1. Thus, the eigenstates of the three-center LMTO Hamiltonian are correctly represented by G˜ to
third order in E E . The first term in (8) introduces two unphysical poles at z =E i/√3p for eachorbital. It is
ν ν
− ±
a diagonal matrix which is real on the real axis and has no effect on the spectrum of the physical states.
B. Spin-orbit coupling in GF-LMTO
In the Green’s function formalism the perturbation can be introduced through the Dyson equation
G =G +G ΣG (9)
V 0 0 V
where G and G are the unperturbed and perturbed Green’s functions and Σ the energy-dependent self-energy due
0 V
to the perturbation V. For V representing the spin-orbit coupling, Σ is local.
In the second-order representation G = (z H )−1, and therefore G = (z H Σ)−1. This G can be
0 orth V orth V
− − −
constructed from the potential functions by denoting
C =C+Σ (10)
V
and defining the following nondiagonalmatrices:
1 −1
P =√∆[∆+(z C )γ]−1(z C ) = γ+√∆(z C )−1√∆
V V V V
− − √∆ h − i
µL =[∆+γ(z C )]−1√∆=(z C )−1√∆P
V − V − V V
µR =√∆[∆+(z C )γ]−1 =P √∆(z C )−1
V − V V − V
λ =[∆+γ(z C )]−1γ =γ[∆+(z C )γ]−1 (11)
V V V
− −
The relations (4) are then replaced by
1
P˙ =µRµL, λ = (µR)−1P¨(µL)−1. (12)
V V V V −2 V V
A tedious but straightforwardderivation then shows
G =λ +µL(P S)−1µR. (13)
V V V V − V
IftheunperturbedGreen’sfunctioniscalculatedtothird-orderaccuracy,wecanstillproceedfromDyson’sequation,
but the situation is complicated by the first term κ = z˜′′/(2z˜′) in Eq. (8) for G˜ , which modifies the perturbed
0
−
Green’s function G˜ . However, for spin-orbit coupling this modification may be neglected, as we now argue. Let us
V
denote G˜ =κ+G¯ and write down the diagrammatic expansion of G˜ in powers of Σ. It is just a usual expansion
0 0 V
in which each Green’s function line can be either G¯ or κ. Now let us resum all diagram insertions with no external
0
lines and no G¯ lines inside. This gives a renormalized self-energy Σ¯ = (1 Σκ)−1Σ. Taking this into account and
0
−
performing the remaining summations, we find
G˜ =G¯+κ+κΣ¯κ+κΣ¯G¯+G¯Σ¯κ (14)
V
where
G¯ =(1 G¯ Σ¯)−1G¯ . (15)
0 0
−
The quantity Σκ p(E E )Σ is very small for spin-orbit coupling in transition metals: Σ 50 meV, p 0.01
ν
∼ − ∼ ∼
eV−2, E E < 5 eV. Since κ is analytic on the real axis, Σ¯ has no poles there too. Therefore, the poles of G˜
ν V
coincide w−ith th∼e poles of G¯. The latter are just the poles of G¯ which are shifted by the self-energyΣ¯. But Σ¯ differs
0
from Σ by a factor (1 Σκ)−1 which is equal to 1 up to small corrections of order Σκ. Up to similar corrections,the
−
residues of the poles of G˜ are equal to those for G¯. Thus, neglecting all small terms or order Σκ compared to 1, we
V
find G˜ G¯+κ.
V
Substit≈uting the second term in (8) as G¯ in (15), we find the perturbed Green’s function:
0
−1
G˜ =√z˜′ z˜ H √z˜′Σ√z˜′ √z˜′. (16)
V orth
(cid:16) − − (cid:17)
3
where we have dropped the inconsequential term κ. This expression for G˜ is equivalent to
V
G˜ =λ˜ +µ˜L(P˜ S)−1µ˜R (17)
V V V V − V
with redefined potential functions
P˜ =P (z˜), µ˜L =√z˜′µL(z˜), µ˜R =µR(z˜)√z˜′, λ˜ =√z˜′λ (z˜)√z˜′, (18)
V V V V V V V V
where instead of C in (10) we should use
V
C˜ =C+√z˜′Σ√z˜′. (19)
V
The self-energy matrix for spin-orbit coupling within the given l subspace is defined as
ΣSO =ξlσσ′(z) lmσ SLlm′σ′ (20)
h | | i
where the energy dependence of the coupling parameter ξlσσ′(z) comes from the radial wave functions. To second
order in (z E ), the radial function in the LMTO basis is φ (z)=φ +(z E )φ˙ . Therefore, we have
ν νlσ νlσ ν νlσ
− −
ξlσσ′(z)= φνlσ(z)VSO(r)φνlσ′(z)
h | | i
= φνlσ VSO(r)φνlσ′ +(z Eνlσ) φ˙νlσ VSO(r)φνlσ′ +(z Eνlσ′) φνlσ VSO(r)φ˙νlσ′
h | | i − h | | i − h | | i
(cid:2)
+(z Eνlσ)(z Eνlσ′) φ˙νlσ VSO(r)φ˙νlσ′ / [1+plσ(z Eνlσ)2][1+plσ′(z Eνlσ′)2] (21)
− − h | | i − −
(cid:3) p
where the square roots in the denominator come from the normalization of φ (z).
νlσ
An approximate form can be obtained by neglecting the terms of order p(E E )2 when they appear in the
ν
−
perturbation, i. e. approximating √z˜′ 1 in (19) and neglecting the square root in the denominator of (21). We
≈
found that for (Fe Co ) B alloys this approximation captures all the essential physics (see, for example, Fig. 1),
1−x x 2
while the p(E E )2 terms only change K by an average of about 10%.
ν
−
The abovetreatmentofspin-orbitcoupling is extendedto CPAwith no modifications. Some implementation notes
about our CPA code can be found in Ref. 8.
II. RELATION TO SPIN-ORBIT COUPLING ENERGY
When MCA energy K appears in second order in spin-orbit coupling, the anisotropy of the expectation value of
the SO operator ∆E = V V is approximately equal to 2K.9 Here we transcribe this relation in terms
SO SO x SO z
h i −h i
of Green’s functions.
The single-particle energy is by definition
1
E = ImTr zG˜ dz (22)
sp −π Z V
The second-order term from perturbation theory is
1 1
E(2) = ImTr zG˜ ΣG˜ ΣG˜ dz = ImTr zG˜2ΣG˜ Σdz (23)
sp −π Z 0 0 0 −π Z 0 0
The expectation value of V , with the radial integral included in Σ, in the same order is
SO
1 1
V = ImTr ΣG˜dz Tr ΣG˜ ΣG˜ dz. (24)
h SOi −π Z ≈−π Z 0 0
AnexactquasiparticleGreen’sfunctionwith simple polesofunitresiduesatisfiesdG/dz = G2. ForG˜ this identity
0
−
is satisfied approximately up to a real term of second order in (E E ). Thus, setting dG˜ /dz G˜2, we find
− ν 0 ≈− 0
1 1 d V
E(2) ImTr z (G˜ ΣG˜ Σ)dz+((dΣ/dz))= h SOi +((dΣ/dz)) (25)
sp ≈ π Z 2dz 0 0 2
where the first term was integrated by parts, and ((dΣ/dz)) denotes the terms with the energy derivative of Σ. For
substitutional alloys treated in CPA, the Green’s function does not have a purely quasiparticle structure, and in
general the relation dG/dz = G2 is violated due to the energy dependence of the coherent potential. This energy
−
dependence contributes additional terms to the right side of (25). However, here we are dealing with relatively weak
substitutionalFe/Codisorder,andtheelectronicstructurelargelypreservesits quasiparticlecharacter. Therefore,we
(2)
may expect the relation E V /2 to hold approximately, which is borne out by calculations. Deviations from
sp SO
≈h i
this relation can also occur near the points of degeneracy close to the Fermi level, where second-order perturbation
theory breaks down. As seen in Fig. 4, this situation appears near x=0.8 where the Fermi level cuts through a pair
of flat, nearly degenerate bands. From Fig. 1 we see the largest difference between K and K at this point.
SO
4
III. SPIN-RESOLVED ORBITAL MOMENT ANISOTROPY
Orbital moments and their dependence on the direction of magnetization are often discussed in connection with
MCA.9–14 In (Fe Co ) B both spin channels contribute comparably to K, and therefore there is no direct relation
1−x x 2
between K and the anisotropy of the total orbital moment. However, the spin-flip terms are small and change
slowly when x is not close to 1 (see Fig. 1). Since the energy dependence of the SO parameters is relatively weak,
the contributions K and K are closely related to the orbital moment anisotropy (OMA) for the states of the
↑↑ ↓↓
corresponding spin. Fig. S1 shows that the concentration-weightedaverage of the spin-down OMA behaves similarly
to K . The spin-up OMA is almost constant except for the Co-rich end where, as seen in Fig. 1, the spin-flip terms
↓↓
also increase substantially. Note that negative OMA for spin-up states corresponds to positive K , but the sign of
↑↑
OMA for spin-down states is the same as that of K . This is because L appears in K with a minus sign, as
↓↓ z ↓↓ SO
h i
reflected in Hund’s third rule.
FIG. S1. Orbital moment anisotropy (OMA) for majority and minority-spin states of Fe and Co. The thick black line shows
theconcentration-weighted average of Fe and Co contributionsfrom theminority-spin states.
The concentration dependence of K and of the majority-spin OMA is weak because the majority-spin 3d bands
↑↑
are fully filled. This feature is typical for Fe, Co, and Ni-based magnetic alloys. Therefore, while K is almost never
proportionaltothetotalOMA,theirderivativeswithrespecttotheconcentrationmaybestronglycorrelatedinmany
magnetic 3d alloys.
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