Handbook of magnetism and advanced magnetic materials

2912 Pages·2007·68.15 MB·English

by  Helmut Kronmüller

Density-functional Theory of Magnetism
Gustav Bihlmayer
Institutfu¨rFestko¨rperforschung,Ju¨lich,Germany
Magnetism in the solid state comprises a large variety of
1 Introduction 1 phenomena that can be characterized by quantities like the
magneticmoment,themagneticorder,theorderingtempera-
2 Density-functional Theory 2
ture, the magnetization direction, and so on. The physical
3 Extensions to ‘Standard DFT’ Approaches 10
concepts, used to describe these quantities on a theoreti-
4 The Magnetic Ground State and Beyond 15
cal basis, are again diverse and include relativistic quantum
Acknowledgments 23
mechanics, statistical theories, and classical electrodynam-
References 23 ics. The magnetic moment of a solid can be obtained, for
example, by quantum-mechanical theories, while ordering
temperatures can be accessed by statistical theories which
use input parameters of quantum-mechanical calculations.
1 INTRODUCTION Exchange interactions have to be described on a quantum-
mechanical basis, but it is usually sufficient to describe
dipole–dipole interactions classically.
The description of many physical properties of condensed
Although there is no single theoretical approach to all
matter involves the theoretical treatment of an inhomoge-
magnetic phenomena, it can be stated that magnetism is
neous electron gas in the potential of atomic nuclei. The
a uniquely quantum-mechanical phenomenon and, in the
ground-state structure, the vibrational and optical proper-
solid state, DFT is the most widespread theoretical method
ties, and the magnetic properties of a solid are determined
used to determine the magnetic properties of matter. Vector-
by the electrons and their interactions, quantum mechani-
spin DFT allows – at least in principle – the access to
cally described by the (many-body) Schro¨dinger equation.
the spin magnetic moment and the magnetic order of the
Wavefunction–basedapproximations,liketheHartree–Fock
ground state and it can serve to extract other quantities,
method, have a long tradition of dealing with this compli-
like exchange interactions, that can serve as input for other
cated many-body problem. In the last 40 years, density-
theoretical approaches. Although DFT is designed as a
functional theory (DFT) has developed into a successful
ground-statemethod,thatis,allresultsrefertoatemperature
alternativetothesemethodsanditseemstobecurrentlyone
of zero Kelvin, it can, nevertheless, be used to infer some
ofthemostwidespreadmethodsincomputationalsolid-state
finite-temperature properties of materials. This contribution
theory (Argaman and Makov, 2000). It has applications in
starts with an outline of DFT and describes the successes
diversefieldslikestudyingthepropertiesofdefectsinsolids,
and current limitations of the theory. Some methods to
heterogeneous catalysis, organic and biomolecular reactions,
overcome these limitations, like self-interaction correction
and also magnetism.
(SIC) or orbital polarization (OP), are discussed. Concepts
to formulate spin dynamics based on DFT are presented
HandbookofMagnetismandAdvancedMagneticMaterials.Edited
and strategies to determine the magnetic ground state of a
byHelmutKronmu¨llerandStuartParkin.Volume1:Fundamentals
and Theory.  2007 John Wiley & Sons, Ltd. ISBN: 978-0-470- systemarediscussed.ThisinvolvesalsothemappingofDFT
02217-7. resultsonmodelHamiltonians,whichcanbeusedtodescribe
2 Electrontheoryofmagnetism
other properties – like ordering temperatures – in a certain the form
(cid:1)
approximation. Orbital magnetism, which is particularly
important in low-dimensional systems, is briefly discussed E[n(r)]= v(r)n(r)dr
here and – in more detail – in later chapters. (cid:1) (cid:1)
Mostexamplesgiveninthiscontributionaremetallic sys- +1 n(r)n(r(cid:1))drdr(cid:1)+G[n(r)] (1)
tems, typically transition metals and their alloys and com- 2 |r−r(cid:1)|
pounds. DFT has proved to be particularly valuable in the
wherethefunctionalG[n(r)]containsallothercontributions.
descriptionofthesematerials,whileotherclassesofmagnetic
If we succeed in finding the functional G[n(r)] or a
systems – like magnetic insulators – have been traditionally
good approximation to it, the immediate advantage of DFT
approached with other methods. Modern magnetoelectronic
is that, instead of dealing with the full many-body wave
and spintronic applications involving orbitally ordered sys-
function, (cid:1)(r ,r ,. . .,r ), we can work with the much
tems, low-dimensional structures, and so on, may require 1 2 N
more tractable density, n(r). Although more information is
theoretical approaches beyond those presented here. But for
directly accessible from the wave function than from the
themajorityofmagneticsystemsDFTpresentsagoodstart-
density,
ing point for a theoretical description.
(cid:1) (cid:1)
n(r)= dr . . . dr (cid:1)∗(r,r ,. . .,r )
2 N 2 N
2 DENSITY-FUNCTIONAL THEORY ×(cid:1)(r,r2,. . .,rN) (2)
in DFT many physical quantities, like the structural prop-
In any realistic calculation, the straightforward quantum-
erties or bond strength, can be obtained for large systems,
mechanical treatment of a system of identical particles is
where a many-body wave function would be impossible to
limited to a very small number of particles. This is mainly
access. For example, calculations of the ground-state ener-
due to the appearance of the many-body wave function,
gies for different external potentials, as they result from a
which contains a tremendous amount of information – but variation of the lattice parameters in a periodic solid, allow
typically we are interested only in a limited number of thedeterminationoftheequilibriumlatticeconstant,whichis
physicalobservables.DFTstartsdirectlyfromthedensityof nowadays possible to within a few percents. Early attempts
theparticlesinquestion(inourcaseelectrons)and,bypassing to use the density as a key parameter for calculations of
the troublesome many-body wave function, allows thereby periodic solids were made by Lenz (1932) based on the
the treatment of a large number of particles. In this section, statistical method of Thomas (1927) and Fermi (1928). In
only a short and certainly incomplete review of DFT can this approach, G[n(r)] was consideredto contain the kinetic
be given. For a more detailed account of this theory, the energy density (taken to be proportional to [n(r)](5/3)). In
reader is referred to the review of Jones and Gunnarsson the Thomas–Fermi–Dirac methodG[n(r)] evencontains an
(1989) or to the books of Parr and Yang (1989) and Eschrig exchangeenergydensitytermproposedbyDirac(1930)(pro-
(2003). portionalto[n(r)](1/3)).AlthoughtheThomas–Fermitheory
still has its applications today, it never became useful as a
theoretical method for the prediction of material properties
(Slater and Krutter, 1934).
2.1 The non-spin-polarized version of DFT
The key idea, that made DFT a success, was to extract
from G[n(r)] the kinetic energy T of a noninteracting elec-
0
Hohenberg and Kohn (1964) worked out two central
trongasinitsgroundstatewhichhasthesamedensitydistri-
theoremsthatformthebasisofDFT:ForasystemofN par-
bution, n(r), as the interacting system. In this Kohn–Sham
ticles (e.g., electrons) moving in an external potential v(r)
theory (Kohn and Sham, 1965), a new functional
(caused by, e.g., nuclei) in a nondegenerate ground state, (i)
themany-bodywavefunction(cid:1) andv(r)areuniquelydeter- E [n(r)]=G[n(r)]−T [n(r)] (3)
xc 0
mined by the particle density distribution n(r) and (ii) there
existsanenergyfunctionalofthisdensity,E[n(r)],whichis appears,thatremainstobedetermined.E isamuchsmaller
xc
stationary with respect to variations of the ground-state den- term than G and is called exchange-correlationenergyfunc-
sity. These two theorems allow – at least in principle – the tional,since – aswewillseeinthefollowingtext – without
determination of the ground-state density and energy of an E our energy functional E would yield just the energy in
xc
N-particle system.Extracting the classicalCoulomb interac- the Hartree approximation. If we(cid:2)take into account that par-
tionenergy,suchaHohenberg–Kohnenergyfunctionaltakes ticle conservation, that is, N = n(r)dr, has to be ensured,
Density-functionaltheoryofmagnetism 3
we can formulate the stationarity of E in equation (1) with Instead of n(r) and m(r), we can now introduce a
respect to variations of the ground-state density, n, as hermitian 2×2 spin-density matrix, n(r), with components
(cid:1) defined as
δT n(r(cid:1)) δE
0 +v(r)+ dr(cid:1)+ xc −λ=0 (4)
δn(r) |r−r(cid:1)| δn(r) n (r)=(cid:4)(cid:1)|ψ+(r)ψ (r)|(cid:1)(cid:5) (8)
αβ β α
where the Lagrange parameter λ ensures the particle con- The density matrix can then be decomposed into a scalar
servation. Expressing the kinetic energy of the noninteract- and a vectorial part, corresponding to the particle and spin
ing particles via their wave functions, φi, we can recast density:
equation (4) in the form of an effective single-particle
(cid:9) (cid:10)
Schro¨dinger equation, the Kohn–Sham equation: 1
n(r)= n(r)I+σ ·s(r)
(cid:3) (cid:1) (cid:4) 2
−2(cid:1)m2 ∇2+v(r)+ |rn−(r(cid:1)r)(cid:1)|dr(cid:1)+ δδnE(xrc) φi(r)=(cid:5)iφi(r) = 21(cid:7) snx((rr))++isszy((rr)) snx((rr))−−isszy((rr)) (cid:8) (9)
(5)
w(cid:5)hich has to be solved self-consistently since n(r)= where I is a 2×2 unit matrix and s(r) is the spin density
Ni=1|φi(r)|2.WecanseenowthatwithoutExc equation (5) that differs from the magnetization density by a factor of
reduces to the Hartree equation. Therefore, this last term of −µ .Likewise,apotentialmatrixcorrespondingtothisspin-
B
theHamiltonianiscalledtheexchange-correlationpotential. densitymatrixisdenotedasv(r)andcanbewritteninterms
Although λ was introduced as a Lagrange multiplier and of a scalar potential and magnetic field, B(r):
the (cid:5) ’s also should be strictly interpreted in this way, it is
i
usual to derive from the (cid:5)i’s band structures of a crystal v(r)=v(r)I+µBσ ·B(r) (10)
and use the wave functions φ (r) as approximations to true
i
quasiparticlewavefunctions.Somejustificationwillbegiven In 1972, von Barth and Hedin used these matrices to
belowandcomparisonwithexperimentaldataoftenconfirms extendtheDFTconcepttospin-polarizedsystems(vonBarth
this point of view, but there are also well-known examples, and Hedin, 1972), replacing the scalar quantities by their
where this interpretation leads to significant ‘errors’, like 2×2 matrix counterparts. In this way, they were able to
in the comparison of the band gaps of semiconductors and deriveaspin-polarizedversionoftheKohn–Shamequations:
insulators with band structures derived from these (cid:5)i’s. (cid:3)(cid:11) (cid:1) (cid:12) (cid:4)
(cid:1)2 (cid:6) n (r(cid:1)) δE
− ∇2+ αα dr(cid:1) I+v(r)+ xc
2m |r−r(cid:1)| δn(r)
α
2.2 The density and potential matrix (cid:11) (cid:12) (cid:11) (cid:12)
φ(+)(r) φ(+)(r)
× i =(cid:5) i (11)
DFT allows us – at least in principle – to determine the φ(−)(r) i φ(−)(r)
i i
correct ground-state charge density. For a magnetic system
we are interested not only in n(r) but also in accessing In contrast to the original, non-spin-polarized, version of
the magnetization density, m(r), directly. If ψ (r) is the DFT, here the uniqueness of the potential is not guaranteed
α
field operator for a particle of spin α, the particle and the (Capelle andVignale, 2001), but for most practicalpurposes
magnetization density can be formulated as this poses no problems. We can now write the exchange-
(cid:6) correlation potential also as a 2×2 matrix:
n(r)= (cid:4)(cid:1)|ψ+(r)ψ (r)|(cid:1)(cid:5);
α α δE
α xc =v (r)=v (r)I+µ σ ·B (r) (12)
(cid:6) δn(r) xc xc B xc
m(r)=−µ (cid:4)(cid:1)|ψ+σ ψ (r)|(cid:1)(cid:5) (6)
B β αβ α
α,β for which suitable approximations have to be found. A
solution of equation (11) yields again Kohn–Sham wave
where we introduced the Bohr magneton, µB =(e(cid:1)/2mc), functions,from whichthe densitymatrix canbe determined:
and the Pauli matrices (underlined symbols denote 2×2
matrices): (cid:6)N
n (r)= φ∗α(r)φβ(r) where α,β =(+),(−) (13)
(cid:7) (cid:8) (cid:7) (cid:8) (cid:7) (cid:8) αβ i i
0 1 0 −i 1 0 i=1
σ = ,σ = ,σ =
x 1 0 y i 0 z 0 −1
As in the non-spin-polarized theory, a solution for
(7) equations (11) and (13) has to be obtained self-consistently.
4 Electrontheoryofmagnetism
If the density matrix turns out to be diagonal, at any fcc Fe can be stabilized as precipitate in Cu and was
point in space the magnetic moments are aligned along found to have a noncollinear magnetic structure (Tsunoda,
the same direction (z) and the magnetization is called 1989).
collinear. Otherwise, we speak of noncollinear magnetic Numerically, equation (11) can be solved by expanding
structures.Examples ofcollinearmagnetizationdensitiesare φ (r) in a linear combination of suitable basis functions
i
displayed in Figure 1 for bcc and fcc Fe. Experimentally, χ (r). Then equation (11) transforms into a problem of lin-
j
ear algebra, that is, to find the eigenvalues and eigenvectors
of a matrix. The eigenvectors, that have to be determined,
give the linea(cid:5)r combination coefficients, cij, of the expan-
sion φ (r)= c χ (r). In such an eigenvalue problem,
i j ij j
the computational effort scales in the most general case
with the third power of the number of basis functions.
Compared to the nonmagnetic problem, equation (5), this
number is doubled in equation (11). Therefore, the com-
putational effort for a general, noncollinear calculation is
increased by a factor 8 as compared to the nonmagnetic
calculation.
2.3 Collinear magnetic structures within DFT
Supposing that the potential matrices in equations (10) and
(12)arediagonal(i.e.,themagneticandexchangefieldspoint
in z direction), equation (11) decouples into two equations
of the type of equation (5):
(a)
(cid:7) (cid:8)
(cid:1)2
− ∇2+v (r)+v(r)+B (r)+v(+)(r)
2m Coul z xc
×φ(+)(r)=(cid:5)(+)φ(+)(r)
i i i
(cid:7) (cid:8)
(cid:1)2
− ∇2+v (r)+v(r)−B (r)+v(−)(r)
2m Coul z xc
×φ(−)(r)=(cid:5)(−)φ(−)(r) (14)
i i i
wherev nowdenotestheclassicalCoulombpotentialand
Coul
v(±) the exchange-correlation potential that arises from the
xc
functionalderivativeoftheexchange-correlationenergywith
respect to the spin-up (+) or spin-down (−) part of the
diagonal density matrix.
The spin-up and spin-down part of the density can be
calculated from the wave functions as
(cid:6)
n(+)(r)= w(+)|φ(+)(r)|2 and
(b) i i
i
(cid:6)
Figure 1. Isosurfacesofthemagnetizationdensityofferromagnetic n(−)(r)= w(−)|φ(−)(r)|2 (15)
bcc Fe (a) and a (hypothetical) ferromagnetic fcc Fe with the i i
i
latticeconstantofbulkCu(b).TheisosurfaceinbccFeshowsthe
boundary between regions of positive and negative magnetization
density; in fcc Fe a small but negative magnetization density, mz, where the weight factors, wi, are determined such that the
isplotted.Plotsweregenerated witha programofKokalj(2003). density of states (DOS), N(E), integrated up to the same
Density-functionaltheoryofmagnetism 5
Fermi level, E , yields the correct number of particles: The integral magnetic (spin) moment, M, for a collinear
F
system is then (in units of µ ) simply
B
(cid:1) (cid:1) (cid:1)
N = EF[N(+)(E)+N(−)(E)]dE where M =| m(r)dr|= (n(+)(r)−n(−)(r))dr (20)
(cid:6)
N(±)(E)= δ(E−(cid:5)(±)) (16)
i But what is essentially responsible for the formation of a
i magneticmomentintheDFTframework,thatis,whichterm
has to be included in the exchange-correlation potential to
Inthisway,thetwoequations (14)arecoupledintheself- describe magnetic systems? For a qualitative understanding
consistency cycle. Furthermore, a coupling occurs via the of this problem, it is instructive to develop a simple model
Coulomb potential and the exchange-correlation potential, for magnetism in the next subsection.
vx(±c) =v+B(±) = δExδc[nn((±+))(,r(cid:6)n)(−)] (17) 2.4 A Stoner-like model for magnetism
Consider a simple elemental magnet with a magnetic
depending on the functional form of E . A particularly
xc moment, M, per atom(again inunits of µ ). The exchange-
B
simple form can be derived from the local approximation to correlation B-field, v(+)−v(−), can be expanded in orders
xc xc
Hartree–Fock theory and includes only the exchange part:
of M:
(cid:7) (cid:8)1 (cid:1) (cid:13) (cid:14) v(+)−v(−) =Mv(1)+O(M2) where
Exc =Ex =−32 43π 3 dr (n(+)(r))43 +(n(−)(r))43 xc xMc =(cid:1) ExFc[N(+)(E)−N(−)(E)]dE (21)
(18)
Empirical modifications by a multiplicative factor have
been used to compensate for the missing correlation energy, Thispotentialdifferenceactsasamagneticfieldthatsplits
but modern functionals include reliable estimates for the the eigenvalues, (cid:5)i, according to
correlation, so that no adjustable parameters are needed.
(cid:12)(cid:5) =(cid:5)(+)−(cid:5)(−) ≈M(cid:4)φ |v(1)|φ (cid:5) (22)
Systems that can be described by equation (14) are all i i i i xc i
kinds of magnetic materials that assume a collinear mag-
thatis,itintroducesashiftoftheeigenvaluesof±1MI((cid:5) ).
netic order, for example, ferromagnetic, antiferromagnetic, 2 i
In the Stoner model (Stoner, 1939), it is assumed that
or ferrimagnetic states. Since the two equations (14) can
all states are shifted by ±1MI, where I is an energy-
be solved independently, the computational effort for a 2
independentexchangeintegral, the Stoner parameter. Within
collinear calculation seems to be just twice the effort for
DFT,a generalizedStonerparameterI((cid:5)) canbeintroduced
a nonmagnetic calculation. However, most magnetic calcu-
and calculated (Gunnarsson, 1976).
lations are computationally considerably more demanding
In the Stoner model, the magnetization that results from
since the quantities in question (magnetic moments, energy the splitting of the eigenvalues by ±1MI is given by
differencesbetweenvariousmagneticconfigurations)require 2
(cid:1) (cid:3) (cid:7) (cid:8) (cid:7) (cid:8)(cid:4)
much higher accuracy than what is needed for nonmag- EF 1 1
netic systems. To explore different magnetic orders in a F(M)= N E+ IM −N E− IM dE
2 2
system, unit cells much larger than the chemical unit cell
(23)
are required, for example, antiferromagnetic body-centered andthismagnetization,F(M),hastocorrespondtothemag-
cubic(bcc)chromiumrequiresacalculationwithatleasttwo netic moment M that induces the splitting. Since the DOS
atomsintheunitcell(ascomparedtoone,inanonmagnetic is always positive, F(M) can only grow monotonously with
calculation). M. Furthermore, we see from equation (23) that F(M)=
The ground-state density is a property that can – at least −F(−M)andinthelimitM →±∞itlevelsoffto±N (full
in principle – be obtained exactly in DFT. In the same way, spinpolarization).Fromtheseconditions,wecanderivethat
the magnetization density is a property that can be obtained the equation F(M)=M has the trivial solution M =0 and
directly from spin-polarized DFT: twonontrivialsolutionsif(dF(M)/dM)| >1.Thisleadsto
0
the famous Stoner criterion
(cid:6)
m(r)=−µ φ∗(r)σ φ (r) (19) dF(M)
B α αβ β | =IN(E )>1 (24)
0 F
α,β dM
6 Electrontheoryofmagnetism
Within DFT the Stoner parameter can be calculated
from a given exchange-correlation potential and the wave 4 Paramagnetic
functions.Typically,valuesaround0.4−0.5eVareobtained Spin split
for transition metals and I turns out to be only weakly
material dependent. Therefore, it is mainly the variation V) 2 −1/2 IM
in the DOS at the Fermi level, N(EF), that determines s/e
e
whether a material is magnetic or not. DFT calculations of at 0
the (paramagnetic) DOS of a material already offer a first st
S (
indicationofwhether(ferro)magnetismhastobeexpectedor O +1/2 IM
not. Assuminga parabolicDOS ofthe d-states,this explains D −2
why the magnetic moment is large in the middle of the 3d
transition metals (Mn) and gets smaller for elements with
−4
almostfilledoralmostemptyd-band.Thisbehaviorissimilar
to Hund’s first rule, and can be observed well for thin −8 −6 −4 −2 0 2
magnetic films (See also Magnetism of Low-dimensional (a) E − E (eV)
F
Systems: Theory, Volume 1).
Of course, the self-consistent calculation with spin-
polarized DFT will give a much more accurate estimate 4 Magnetic
of the magnetic moments than this model. But a com- Spin split
parison between the predictions of the Stoner-like model
and the actual DFT calculation of the magnetic system V) 2
e
(Figure 2) for bcc Fe shows that our simple Stoner-like s/
e
modelisnotsofarfromaself-consistent,spin-polarizedDFT at 0
solution. S (st
An exchange-correlation potential that gives a reasonable O
estimate for the (generalized) Stoner parameter, I, can be D −2
expected to also give reasonable magnetizations. Therefore,
evenrelativelysimple exchangecorrelationenergyfunction-
−4
als, like given in equation (18), describe magnetic trends in
DFT qualitatively correct, provided the nonmagnetic DOS −8 −6 −4 −2 0 2
is also described correctly. Properties that are required for (b) E − E (eV)
F
a (non)spin-polarized exchange correlation potential will be
discussed in the next subsection. Figure 2. (a)ParamagneticDOSofbccFe(black,fullline)shifted
by±1IM.TheresultingDOS(dashedline)iscompared(b)tothe
2
result of a spin-polarized DFT calculation (gray, full line). The
2.5 Exchange-correlation energy functionals spin-downDOSare plottedwithnegativevalues.
One of the first interpretations of the term, which emerged correlation function, g(r,r(cid:1)) (Kohn and Vashista, 1983):
as the exchange-correlation potential in DFT, was given by (cid:1)
1
SlaterandKrutter(1934)inthecontextoftheThomas–Fermi n (r,r(cid:1))=n(r(cid:1)) dλ[g (r,r(cid:1),λ)−1]
xc n
method and later in connection with the Hartree–Fock 0
method (Slater, 1951). Essentially, it describes the interac- ≡n(r(cid:1))h(r,r(cid:1)) (25)
tion of a particle with the ‘hole’ that is created by its own
presence in the gas of the other particles. This means that Here, g (r,r(cid:1),λ) is the correlation function of a system
n
the probability of finding an electron at a position r reduces ofchargedparticleswherethe Coulombinteractionis scaled
the probability of finding another electron at a position r(cid:1) by a factor λ and a λ-dependent potential has been added,
nearby, depending of course alsoon the spin of the two par- so that the density, n(r), is independent of λ. Additionally,
ticles (therefore, in the Hartree–Fock method this hole has the so-called hole function, h(r,r(cid:1)), was introduced. The
been given the name ‘exchange hole’). exchange-correlation energy can then be written as
To derive some properties of the exchange-correlation (cid:1) (cid:1)
1 1
energy functional, it is useful to write this ‘hole’ (exchange- E [n(r)]= drn(r) dr(cid:1) n (r,r(cid:1)) (26)
correlation hole in DFT), n , in terms of a two-particle xc 2 |r−r(cid:1)| xc
xc
Density-functionaltheoryofmagnetism 7
Although the exchange-correlation hole can be very com- functionals are restricted to the form
(cid:1)
plicated in shape, it was soon realized that only its radial
dependence enters in the exchange-correlation energy (Gun- E [n(+)(r),n(−)(r)]= drn(r)ε (n(+)(r),n(−)(r)) (29)
xc xc
narsson, Jonson and Lundqvist, 1979). This means that, in
practice, Exc is rather insensitive to details of shape of nxc. (local spin-density approximation, LSDA) and possibly
Some properties of the exchange-correlation hole can be also containing the gradients of n(±)(r) (GGA). There
derivedfromthedefinitionviathecorrelationfunctiong,for are also formulations including the kinetic energy den-
example,thereisasumrulewhichstatesthatnxccorresponds sity, τ(±)(r), (meta-GGA) and other hybrid schemes (Tao,
exactly to one electron, that is, that Perdew, Staroverov and Scuseria, 2003), which will not be
(cid:1) discussedhere sinceextensivetestsonmagneticsystemsare
dr(cid:1)n (r,r(cid:1))=−1 (27) not yet available. But all these schemes work basically with
xc ‘spin-up’ and ‘spin-down’ quantities, and a study that went
beyond this approximation (proposing a spin angular gradi-
has to be fulfilled. Such relations can guide the construction ent approximation to DFT) indicated rather small effects on
of exchange-correlation functionals or help in judging the thecalculatedpropertiesascomparedtothecollinearversion
validity of existing approximations to Exc. of Exc (Katsnelson and Antropov, 2003).
One of the big surprises in the early days of DFT was Earlycalculationsofelementalferromagnetsdemonstrated
certainly the fact that even a simple exchange-correlation that LSDA is rather successful in the prediction of the
functionalliketheLDAleadstorelativelyconvincingresults. (spin) magnetic moments. The increase of the magnetic
The LDA starts from the limit of the homogeneous electron moment with decreasing coordination (e.g., at surfaces or
gas, assuming E rather as a function than as a functional in thin films) is well described by DFT and chemical
xc
of n(r). Its success can now be explained by the fact trends similar to Hund’s first rule can be observed in lower
that the exchange-correlation hole in the LDA is of the dimensions. Some examples are presented in Magnetism
form of Low-dimensional Systems: Theory, Volume 1; some
results for bulk systems are given in Table 1. It can be
nLDA(r,r(cid:1))=n(r(cid:1))h (|r−r(cid:1)|;n(r(cid:1))) (28) seen that spin polarization leads to a significant expansion
xc 0 of the lattice constant not only for Fe and Co but also
for Mn and several magnetic alloys. This magnetovolume
where h (|r−r(cid:1)|;n) is the hole function of a uniform inter-
0 effect can be explained by the exchange splitting of the
acting electron gas of density n. For a uniform density,
d-band, as observed in Figure 2 for Fe: It leads to an
this exchange-correlation hole satisfies equation (27). For a
enhanced population of antibonding majority-spin states
nonuniform density, the sum rule should be at least approx- (at the top of the band) and a reduced population of
imately fulfilled and Gunnarsson and Lundqvist (1976) bonding and nonbonding states (bottom and middle of the
showed that in LDA this is, on average, the case. This, band) in the minority spin channel. The weaker bonding
togetherwiththe factthatExc dependsonlyonthespherical leads to an expansion of the lattice and also reduces the
average of nxc, is mainly responsible for the success of the bulk modulus of these metals (Janak and Williams, 1976).
LDA. While spin polarization improves the lattice constant as
Also modern, spin-dependent exchange-correlation func- compared to non-spin-polarized LDA, for 3d metals LSDA
tionalsincludinggradientcorrectionsareconstructedinsuch still gives smaller values than observed experimentally. The
a form that they fulfill certain conditions that are known use of gradient corrections and a correct treatment of the
exactly in different limits (like high or low density, con- nonspherical charge distribution in the solid finally leads to
stantorslowlyvaryingdensity, etc.).Inthis way,exchange- reliable lattice constants for these transition metals (Asato
correlationpotentialsareimprovedonaparameter-freebasis. etal., 1999).
Alternatively,thefunctionals(orpartsofthefunctionals,e.g. Here, one has to note that the LSDA results for the mag-
thecorrelationenergy)canbefittedtonumericalresultsfrom netic moments in Table 1 were obtained from calculations
many-bodycalculations.Anotherstrategy – oftenusedinthe assuming the experimental lattice constants of the elements.
chemical literature – is to adjust the functional to yield best Attempts to theoretically describe both the magnetic and
results (like bond length, dissociation energies, etc.) for a structural properties of iron showed that LSDA predicts an
given set of systems. fcc ground state with an almost vanishing magnetic moment
Although the exchange-correlation potential of spin- (Wang,KleinandKrakauer,1985).Forthissystem,itturned
dependent DFT was first formulated in terms of 2×2 spin out that only inclusion of gradient corrections can save the
matrices, almost all practically used exchange-correlation situation and GGA calculations finally correctly predicted a
8 Electrontheoryofmagnetism
Table1.Magnetic moments (in µ per atom) and lattice constants (in A˚) of ferromagnetic elements
B
in the bulk. The experimentally determined total magnetization, M , consists of spin and orbital
tot
moment contributions. The LDA (non-spin-polarized) and LSDA (spin-polarized) results for Fe, Co,
andNiaretakenfromMoruzzi,JanakandWilliams(1978);theGGAvaluesofthemagneticmoments,
from Shallcross, Kissavos, Meded and Ruban (2005) where experimental values are also quoted. The
calculatedGddataisfromKurz,BihlmayerandBlu¨gel(2002)andtheexperimentaloneisfromWhite
etal. (1975). The GGA values of the lattice constants of Fe and Ni are from Asato etal. (1999) and
theColatticeconstantisfromBattocletti,Ebertand Akai(1996).
Property Source Fe(bcc) Co(fcc) Ni(fcc) Gd(hcp)
M LSDA 2.15 1.56 0.59 7.63
spin
M GGA 2.22 1.62 0.62 7.65
spin
M Experiment 2.12 1.57 0.55
spin
M Experiment 2.22 1.71 0.61 7.63
tot
a LDA 2.73 3.41 3.47
0
a LSDA 2.76 3.47 3.47 3.58
0
a GGA 2.83 3.55 3.55 3.69
0
a Experiment 2.86 3.53 3.53 3.63
0
bcc ground state structure with a magnetic moment that is notice that the direction of this magnetization with respect
slightly larger than the experimental one (Asada and Ter- to the lattice did not enter in the formalism and we relied
akura, 1992). on the assumption that the total energy is invariant with
As noted above, for 3d transition metals GGA improves respect to a uniform rotation of the magnetization direction.
thelatticeconstantsand,thereby,inmostcasesalsothemag- This was implicitly assumed when we arbitrarily (or, better,
netic moments. This success of the GGA is contrasted here for convenience) selected in equation (14) the z direction
with the case of Cr, where up to now no satisfactory agree- as global magnetization axis. Indeed, in the absence of
ment with experimental results was obtained: while LSDA an external B-field (or in its presence, as long as it is
calculations of antiferromagnetic Cr at the experimental lat- orientedinthe z direction)this implies nolossingenerality,
tice constant give a magnetic moment in good agreement if the interactions included in v are isotropic in space.
xc
with experimental data (0.5−0.6µ ), and also a slight sta- If we start from a Schro¨dinger–Pauli-like theory, there is
B
bilization of the experimentally observed incommensurate indeed no term that could couple the spin space to the
sinusoidalmodulationoftheantiferromagneticstructurewas lattice. The dipole–dipole interaction, which can introduce
predicted (Hirai, 1998), calculations at the lattice constant amagneticanisotropy,resultsfromarelativistictwo-particle
determinedwithLSDA(whichis3%toosmall)yieldanon- term and is not contained in LSDA or GGA (but it can
magneticgroundstate.GGAcalculations,ontheotherhand, be added as a classical magnetostatic term ‘by hand’, if
give a reasonable lattice constant but the magnetic moment the magnetic moment is known). However, the presence of
is more than 60% too large (Singh and Ashkenazi, 1992). an orbital moment can introduce a coupling between spin
Furthermore, theunmodulatedantiferromagneticstructureof and lattice by the spin-orbit coupling term, which follows
Cr seems to be stabilized by GGA, whereas a modulated from the single-particle Dirac equation. The formation of
structureisobservedexperimentally(Hafner,Spisˇa´k,Lorenz orbital moments in DFT will be discussed in the next
and Hafner, 2002). subsection.
Despite these critical remarks on the success of DFT in
describing Fe and Cr (and much more could be said on Mn
(Hobbs and Hafner, 2003)), it should be emphasized that 2.6 Orbital magnetism
DFT is an excellent theory for the prediction of magnetic
properties of materials – despite some limitations. As the The magnetization density and the magnetization,
vast amount of literature in this field shows, the area of equation (19), are clearly a consequence of the imbal-
applicability of this method is constantly expanding. Recent ance of electrons with spin-up or spin-down and, therefore,
examples from the field of magnetism in low dimensions the quantity defined in equation (20) is called spin moment.
can be found in Magnetism of Low-dimensional Systems: From atomic physics we know that the total magnetic
Theory, Volume 1. moment is a sum of spin and orbital contributions,
Up to now, we focused on the spin density and the M =M +M . The orbital moment results, in a
tot spin orb
resulting, in most cases collinear, magnetization. We should classical picture, from the orbital motion of the electron
Density-functionaltheoryofmagnetism 9
around the nucleus. Compared to the situation in a free Adding this term to equation (11) will destroy the decou-
atom, where M can be even larger than M , in a solid plingofspin-upand-downequationslikeinequation (14).It
orb spin
thismotionisrestrictedbythecrystalfieldthatquenchesthe alsoinvalidatesthe aforementionedassumptionthatthe total
orbital moment. In bulk samples, small moments (typically energyisnotaffectedbyauniformrotationofthespindirec-
0.1–0.2µ ) can be found (compare Table 1). tions:thecrystalfieldinasolidactsontheorbitalmotionof
B
DFT in the known LSDA or GGA formulations provides theelectronsothatthereisapreferentialplaneforthisorbital
notermthatcouldleadtotheformationofanorbitalmoment motion. Therefore, a total energy difference arises when the
in the presence of a crystal field. In an atomic LSDA solidis magnetizedintwodifferentdirections (Sto¨hr, 1999).
calculation of scandium (3d1 configuration), all d-levels This difference, the magnetocrystalline anisotropy energy
of a given spin (m =−2,−1,. . .,2) are energetically (MCA) energy, is small for bulk systems with high sym-
l
degenerate if no external magnetic field is present. A small metry, for example, cubic crystals like Fe or Ni. It is larger
crystal field would immediately lead to the formation of for crystals with a unique crystallographic axis, like hexag-
linearcombinationsoftheselevels(e.g.,dxy ordx2−y2)which onal Co. But for lower-dimensional systems, thin films, or
carrynoorbitalmoment.Current-andspin-densityfunctional atomic wires, the MCA will essentially determine the mag-
theory (Vignale and Rasolt, 1988) would provide a natural neticproperties,especiallyatfinitetemperatures(Bihlmayer,
starting point for the description of orbital magnetism, but 2005).ExamplesofcalculationsoftheMCAwithinDFTcan
so far the success is limited. be found in Wu and Freeman (1999). The MCA energy is
As noted above, the Dirac equation provides a term that alsodiscussedinTheoryofMagnetocrystallineAnisotropy
couples the spin to the orbital moment and, thereby, coun- and Magnetoelasticity in Transition-metal Systems, Vol-
teracts the influence of the crystal field and leads to the ume 1.
formation of an orbital moment. For a deep and funda- Theorbitalmagnetizationcanbedefinedinanalogytothe
mental discussion of relativistic effects in DFT, we refer spin magnetization, equation (19), now expressed in single-
the reader to a paper by Jansen (1988) and the review of particle wave functions φ :
i
Staunton (1994). Here, we take a simpler approach: inter- (cid:6)
preting the Kohn–Sham equation or its spin-polarized form, morb(r)=−µ (cid:4)φ |r×v|φ (cid:5) (31)
B i i
equation (11),likeasingle-particleSchro¨dingerequation,we
i
cancomparethisequationwithatwo-componentapproxima-
tiontotheDiracequationfromrelativisticquantummechan- At a certain atom ν, the orbital moment Morb can then be
ν
ics (Bethe and Salpeter, 1977). This Pauli equation has the obtainedbyanintegration(denoted(cid:4)(cid:5) )inaspherecentered
ν
sameformastheSchro¨dingerequation,butadditionallysev- around this atom:
eral terms, like the Darwin or mass-velocity term, appear as (cid:6)
aconsequenceoftherelativistictreatment.Ifthesetermsare Morb =−µ (cid:4)φ |L|φ (cid:5) (32)
ν B i i ν
now included in the Kohn–Sham equation, we can hope to
i
improve the description of heavier elements, without having
to modify the exchange-correlation potential; for example, Although this definition of the orbital moment poses
thescalar-relativisticapproximationtakesoverfromthePauli no difficulties in periodic solids, we note here that the
equation these terms that are diagonal in spin and prove to evaluation of the total orbital moment of a periodic crystal
be essential for the description of 5d elements. is more involved (Thonhauser, Ceresoli, Vanderbilt and
AnothertermofthePauliequation,thespin-orbitcoupling Resta, 2005). As will be shown in later chapters, relativistic
term, couples up- and down-spin and also provides a DFT calculations successfully predict the chemical trends
mechanism that leads to orbital polarization, that is, the (similar to Hund’s second and third rules), the changes of
formation of an orbital moment. The electron, traveling on the orbital moment with the coordination of the atom or
a classical trajectory around the nucleus, experiences the the dimensionality of the system, and the variation of the
electricfield(fromthescreenednucleus)asamagneticfield. orbital moment with the magnetization direction. It should
This field couples to the magnetic (spin) moment of the be noted that in most cases the atomic orbital moments
electron and, thus, leads to a preferential orientation of the and also the MCA energies, obtained in DFT calculations,
spintotheorbitalmotion.Usingtheorbitalmomentoperator are too small compared to experiment; for example, the
L=r×v, we can write the spin-orbit coupling term in the calculated orbital moments of Fe, Co, and Ni in LSDA are
vicinity of a nucleus with a radial potential v(r) as 0.05, 0.08, and 0.05µ , respectively (Beiden etal., 1998).
B
Compared to the experimental values (Table 1, difference
1dv(r) between total and spin moment) the values of Fe and Co
H = (σ ·L) (30)
so
r dr are about 50% too small, for Ni the value is surprisingly