Download Physical origins and theoretical models of magnetic anisotropy

24.1
8
P.Bruno
lnstitut d'Electronique Fondamentale, CNRS UA 022
B3t. 220, Universite Paris-Sud, F-91405 Orsay, France
1
Introduction
The prirnary properly of a ferromagnet, sucli as Fe, CO, or Nil is the appearance of a
sporit,aneoi1s rnagriet izaI,iori M below the Curie temperature TC. The energy involved by
this sponiancrieous IxealtiIig of synirnetry is of tlie order ol ]igijlb/atoIn M 0.1 eV/atom.
The nic~c~lianisrti
resporisi1)le for the appearance of ferrornagnetisrtl has hren recognized
1)y 11eisc-nl)erg Io Iw Ilie P u d i jwinciple, which prevents two electrons or parallel spins
t o occupy the same orbital state, so that, the eflectzee Coulomb repulsion for a pair of
elc~troiiswit 11 parallel spins is wcaltcr than for antiparallel spins; this is known as the
c .rch n n gc in t c r(i&o 11.
For a t heoret,ical description of the basic properLies of ferromagnetic materials, it
is suffirieiit l o use 1?07?-7'(hLfi?hticquantum mechanics. However, the spin is introduced
Iic.re i n an n d hoc manner, so that there is absolute €reedom in the choice of t h e spinquaiit izatiori axis; i n otlier words, non-relativistic qiiantum mechanics leads to a descript ion of ferrorriagnet isin i n wliicli tlie free energy of the system is independent of tlie
direction of the magnetization (it is said to be isotropic). This is in contradiction with
experience, wliic.11 tells 11s that, the xnagIiet,ization generally lies in some prefewd dzreclions with rcspect to tlie crystalline axes and/or t o the external shape of the body: this
propcrty is known as Ihe mcignetic nnisolropy.
Thc, ciicrgy involved in rotating tlie magnetization from a direction of low energy
(easy axis) towards a one of high energy (hard axis) is typically of the order of lo-'
cV/atoIn. This anisofropy energy is thus a very small correction to the total
lo
iiiagnrtic energy; it actually arises from relntivzstzc corrections to the Hamiltonian, which
lmalt (he rotat,ional invariance with respect ot the spin quantization axis: these are the
24.2
dzpole-dapole interactzon and the spin-orbat coupleng.
Expressed in units of magnetic field, the magnetic anisotropy is of the ordcr of 0.1 to
100 kOe, i.e. of the order of magnetic fields used in experimental situations. Thus, it appears clearly that theses relativistic corrections should play an essential r81e; in particular,
tlie magnetic anisotropy is a key property for applicatiorls where the rnagnetizat,ion must
he pinned in a given direction, such as permanent magnets and media for magnetic storage
of inforrnation. On a more fundamental point of view, the dipole-dipole interaction arid
t h e spin-orbi t coupling are necessary to explain the very existence of ferrornagrietism in
two-tiirnensional systems such as ultratliin films: indeed, according to the Mermin-Wagner
theorcm, two-dimensional systems with short-range, isotropic exchange interactions, h t
without clipole-dipole or spin-orbit intcractions, cannot sustain ferromagnetic order a t
non-zero temperature.
The present Chapter is devoted to the discussion of the physicdl origins and theoretical models of magnetic anisotropy. It is organized as follows: in Sec. 2, we present
t lie pheriornenological description of magnetic anisotropy, at, A macroscopic levcl, with
emphasis on sq'tnrii~tryconsiderations; Secs, 3 and 4 treat,, at the microscopic level, the
rriagriet ic. ariisot ropy arising, respectively, from the clipol~-dipoleinteractions, anti from
t h r spin-orhit coiipling.
Slwcial ertiphasis will be given to the magnetic anisotropy of inteiface atoms in
tiltrathin lilrns and multilayers, which is much larger than the onr of hulk atoilis, ancl is
c-iirrmt Iy of very strong interest, from the fundamental point of view, as well as fol storagc
applications. Altliougli rare earth metals and rare earth-transition metals coinpotnicls
liavr w r y large magnetic anisotropies, they will not be discussed lirrc; rather, we will
concciitrate on tiansition metals (Fe, CO, Ni), in which tlie magiictic moment is carried
bj. t hc delocalized 3d electrons (itinerant ferromagnetism). Mk notc in passiiig lliat the
5pin-ohii coupling is also I'eSpOnSibk for ot1ic.r properties of strong I'untlament,al and
t cclinological interest, siich as the magneto-optical effects, the magnetic. circular clicliroism,
or the extraordinary Hall effect; they will not, be discussed hcre. Throughout, this C h p t e r ,
( .g.s. units will he used.
2
2.1
Phenomenology of Magnetic Anisotropy
Thermodynamic description
Let 11s consicler a ferrorriagnet,ic body of magnetization M = M O M , subrnitt,ed t o a
uni form external field H. In order to unambiguously separate the magnetic anisotropy
from energy contributions related to the exchange interaction (which are not of interest
here), we rcstrict ourselves to situations where the unit vector O M of the magnetization
dircction is uniform t,hroughout the sample; in the following,
will be describcd cithcr
by its coinponeiits ( a l , a 2 ,a3) with af a: a; = 1, or by the polar angles 0 and #I,
defined in thc usual manner. Also, we consider only temperatures well below the Curic
temperature, so that magnetization fluctuations can bc neglected.
T h e free energy density F ( T ,Ad, O M ,E ) is thus a function of the temperature T,the
rriagnetizatiori magnitude A4, the tnagnetization direction O M ,and the shain tensor E .
One should realize that, this free eiiergy is not very convenient to use, essentially because
M is not an externally controlled parameter. Actually, in a typical experimenlal situation,
t h e magnetization is rotated from a direction to another one by rotating an external field of
given magnitude; the magnitude of the magnetization M changes as the latter is rotated.
+ +
24.3
Clearly, one needs t o change the free energy F ( M , . . .) for a thermodynamic fuiiction
having the external field as a natural variable. The appropriate Legendre transformation
has been discussed by various authors [l];the convenient thermodynamic potential G' is
given by
C(T,H M ,Q M , E ) F - H M M ,
(1)
where H M is the projection of the external field along the magnetization direction O M .
Thus, appart from the temperature and the strain tensor, the natural variables of G are
IIhf and Q M = (8,4), and the corresponding partial derivatives are
Except cxplicitely specilicd, tlic strain tensor E and the cxternal field component HA,
along Q M will
talcen equal to zero in the following, and the thermodynamic potential
will be noted G'(W,w); for simplification, G will be refered to as the energy of the system.
Not P that taltirig 11~4= 0 corresponds to experirriental situations where t h e external field
is i)~rperidicularto tlie magnetization; this is sornehow idealized, for i n usual cases a
IIOII-zero
field H M is necessary to maintain the system in a single-domain state.
11 is also important to keep i n mind that, as already mentioned, the magnetization
magnitudP A4 itself is anisotropic, i.e. that it depends on i - 2 ~ . The anisotropy of the
magnetization is re la td to the dependence or the anisotropy energy on H M by the Maxwell
relations,
2.2
S h a p e versus crystalline anisotropy
As we have already mentioned in the Introduction, the energy depends the orientation
Qhf of the magnetization ( i ) with respect to the crystalline axes of the ferromagnetic
body, a11d (ii) with rcspect t o its external shape.
In order t o unambiguously establish this distinction, let us perform the following
~ r r l n n 8 e ne.cpr7immt: we take a ferromagnetic material witli a cubic crystalline structure
and cut two samples, (a) a spherical one, and (b) a thin plate with the normal parallel to
the [OOl] axis, as depicted on Fig. 1. The spherical sample is easily magnetized along the
[OOl] and [loo] directions (easy axes), whereas a larger field is needed to magnetize it along
the [loll direction (hard axis); since the shape of the sample is isotropic, the observed
anisotropy implies that the energy depends on the orientation of
with respect to
the crystallinr czzes; this is ltnown as the muynetocrystulline anisotropy. On the other
hand, for tlie plate-shaped sample, different magnetization curves are reported for the
[loo] and [ O O l ] directions, which are respectively parallel and perpendicular to the plane;
since these two axes are crystallographically equivalent, this indicates that the energy also
depends on the orientat,ion of i l M with respect, to the shape of the sample. Thus the total
anisotropy energy niay be expressed as
G ( a M )= Gcryst.(QM) -k Gshape(OM)
*
(6)
24.4
Figure 1: (a) Magnetization curves along the [loo], [OOl], and [loll axes, for a spherical sample.
(b) Magnetization curves along the [loo] and [OOl] axes, for plate-shaped sample.
It is clear that the first, term is an intrinsic contribution, depending only on the ferromagnetic material under consideration, whereas the second one is essentially of geometric
character. In our gedanken experiment, we selected situations where the anisotropy arises
entirely from one of these two contributions; however, in usual cases, bot,h shape and
magnetocrystalline anisotropy contributions are present. Thc general method to separate
them will be discussed in Sec. 3.1.
The shape anisotropy arises entirely from the dipole-dipole interactions, while the
magnetocrystalline anisotropy arises essentially from the spin-orbit coupling, but also, t o
a lesser extent, from the dipolar interactions.
2.3
Symmetry considerations
The general form of GCryst.(i2M) for a given crystalline structurc can be found by using
some symmetry arguments. First, the invariance of the Hamiltonian with respect to
tinie reversal implies that the expression of Gcryst.(i2M) must remain unchanged if i 2 ~
is replaced by --OM. T h e most convenient way to express the anisotropy energy is to
expand it in sperical harmonics:
Gcryst .
=
1 even m=-l
Another possibility is to expand the anisotropy energy in successive powers of the components (01, NZ, 0:3) of O M :
24.5
G c r y s t . ( a d = bo(ffA,f)
+
43(HM)%%
2-3
4-
(8)
& k i ( H , v i ) ~ , a ; ~ k Q " -I-.
*.
%,j3k,l
In Eqs. (7,8), only those terms that are even in f l (i.e.
~ compatible with the time-reversal
symmetry) liave been included. Although the expansions (7) and (8) are equivalent,
the splierical harmonics prescnt t he advantage of forming a complete set of orthonormal
functions; unfortunatcly, the tradition has established some expressions for G ( S ~ M
which
)
(lo riot possess this property. Similarly, the magnetization can be expanded in terms of,
e.g., sptierical harnionics,
1 even m=-l
Tlie anisotropies of t,lie energy and of the magnetization are related by t h e Maxwell
Exprric~ic~~
sliows that such expalisions converge rapidely with increasing orcler, SO that
a few trrrris are eriougli t,o describe acciirately the rr~agnetocrystallirie anisotropy. Tlie
explanai ion for this rapid coiivergence will appear clearly in Sec. 4.
The cryst alliiie syrnriietry irnposes some relationships between the coefficients of
given ortlcxr, thcreby reducing tlie number of imdcpendcnt parameters. For example, in
cubic systems such as Fe and Ni, terms of order 2 are forbidden and the first non-vanishing
coiitribution to tlie crystalline anisotropy is of order 4. The usual expression for the
anisotropy of cubic systems is
Gcryst,(aA,f) I<"
krl(a;U; t
CYia:) Jir201a~a,
2 2 2
~ ( n h =f )hf0
+
+
+
+
+
+
t .;a; +
t ...
n/ll(nfa; +a&: + a+;) + II/l.La,a2a,
n/f:<(LY;..;
+;.;a + a3a1)t - ,
(11)
&(a;LY;
2
2
2 2
2
2
(1'2)
'
with llic coordinates axes taken along the cubic axes. For systems with licp structure,
like CO, thc usual cxprcssion of the anisotropy is
+ KI sin20 + I<zsin40 ( K 3 + KJ cos(64)) sin6U + - .
n / l ( n ~=~hfo) t 11/11sin2 0 + h/12sin48 + (M3 + h!ficos(6$))sin6 0 + . . . ,
GcrJrst.(S2M) = 110'
*
(13)
(14)
wlicre 4 aiid 0 arc takcn with rcspect to thc U and c axes, respectively. Note that the
traditional constarits, K1, It'2, etc., are somehow misleading; for instance, Ii'l is a constant
of' order 4 i r i cubir systems, and of order 2 in hexagonal systems.
The values of' the anisotropy constants of Fe, CO, and Ni are given in Table 1. The
casy axcs of Fe and Ni arc respectively the [lo01 and [111] directions, while that of Go
is along tlie r axis. It, is worth noting that Ihe anisotropy of hcp CO, which lias a lower
syrnmetry, is one order of magnitude larger than that of Fe and Ni, which have a cubic
symmetry. Also, we can rernarlc that the sign of M
I is the opposit of that of I { i , which
means that the magnetization is larger in tlie easy directions than in the hard ones. These
points will be interpreted in Sec. 4.
24.6
Table 1: Anisotropy constants of Fe, CO, and Ni, at 7' = 4.2 I(. (")Ref. [2]; (b)Ref. [3]; (')Ref.
[4];
(d)Kef. [5]
2.4
Volume versus interface anisotropy
So far, we Iiave implicitly supposed that the systctn uiidcr considerat,iori is large enough
for the surlacc contributions to the energy to hc negligible. This is liowcwr riot tlie
case i n systems of sinall dimensions, such as ultrathin films; in siicli systems, tlir total
therrnody~iarnicpotential must he written as the sum of a volume t,errn, ancl of a surlizce
(or interface) contribution, i.e.
is the energy per unit volume (this the onc we have discussed so far), and
where G"(n,)
G'(Ob1) is the energy per unit interfacial area. The latter usually depends tlic iriatcrisls
in contact a t the interface and on tlie crystalline orientation of the latter.
It was first pointed out by Ndel [ti]that the atoms located near an interface have a
tliffcrent, environment as compared to bulk atoms, and that they give additional contributions to the magnetic anisotropy. In particular, since the symmetry of an interface is
often lowcr than that of the bulk, anisotropy t e r m that are forbidden i n the bulk may
be present at an interface.
The expression of the surface contribution to t h e magnetocrystalline energy is
GZryst (a,) = ICf sin2 0 + (I<:
+ K~scos(4$))sin4 0 + . . . ,
(16)
for a cubic (001) surface (tetragonal symmetry),
+
G & ~ ~ ~ . ( Q=
M() I C ~ K:" cos(24))sin2 B
+ ... ,
(17)
for a cubic (110) surface (orthorhombic symmetry), and
G ' & y s t . ( Q ~=
) Kf sin2 0
+ ICt sin40 + (I<: +:'11
cos(6$)) sin60
+ .. . ,
(18)
24.7
c
-5
5 -5
0
H
0
5
hoe)
Figure 2: Magnetization curves at, T = 10 K 01 Au/Co(0001)/Au(lll) sandwiches, measured
perpendicularly (left) or parallel (right) to the film plane, for various values of the CO thickness
tco: the easy axis is perpendicular to the plane tCo < t , x 12 A. From Ref.[7]
for a cubic (111) or an hcp (0001) surface (hexagonal symmetry). In t h e above equations,
the angle B is measured with respect to the surface normal.
In practice, only the first terms (order 2) are taken into account. Their order of
magnitude is typically of 0.1 to 1 erg.cm-2; in terms of microscopic units, this amounts t o
about
eV.(interface atom)-’, which is considerably larger than the anisotropy
to
of bulk atoms. The sign of the surface rnagnetocrystalline anisotropy may be positive or
negative, depending on the interface under consideration.
The situation where Kf is positive is of particular interest in ultrathin films: indeed, as we will see in Sec. 3, the volume anisotropy of films is dominated by the shape
contribution which favors a in-plane orientation of the magnetization; the latter is competed by the surface anisotropy which favors a perpendicular magnetization for K,” > 0.
Thus, at large thickness, the bulk term dominates and the magnetization lies in the plane,
whereas the relative weight of the surface terms increases with decreasing thickness, SO
that eventually, below a. critical thickness of the order of 10
the magnetization becomes
perpendicular to the plane. An exemple of this behavior is shown on Fig. 2. This is of
strong interest for technological applications in magneto-optical disks.
A,
24.8
2.5
S t r a i n - i n d u c e d a n i s o t r o p y and m a g n e t o s t r i c t i o n
In the abovc discussion, we havc talcen the strain tensor E t o be zero. If the latter is not
zero, new energy terms must be considered. Some of them are purely elastic, i.e. they
depend only on the strain tensor E . However, in a magnetized body one also has eIicrgy
terms that depend both on E and on O M : the magneto-elastic energy. In all this Section,
we shall consider only volume terms, and consequently drop the corresponding V indices.
As usual, for small deformations, one can formally expand this energy in powers
of the deformation ancl in spherical harmonics of fl211.1 (or powers of the a;'~).
T h e nonvanishing terms of lowest order are linear with respect to B and quadratic with respect to
the Q,'s. Thus the most general expression for the magncto-elastic energy density is
As for the magnetic anisotropy, the crystalline symmetry imposes some relationshilx
1x4ween the coeIficierits BiJkl. Thus, for cubic systems, the standard expression of the
nisgnrtoelastic energy is:
+
+
c,Ilagll.el.(fiM,E ) = BI(EIIQ;E Z Z ~ ;
+
2 B 2 ( E 1 2 0 1 a2
+
2
€3303)
E23a2a.7
+
+
& 3 1 ~ 3 ~ 1 )' ' ' >
(20)
aiid lor hcp crystals:
As tlic syminetry is generally lowercd uridcr strain, tlic magneto-elastic c n c ~ g ynlay cont ain aiiisoi ropy terms that are forbidden in the unstrained state; for instance, c u l i c crystals uiidrr st rain acyuirc anisotropy terms of order 2. The magneto-elastic consl,anf,s &,
132. c ~ c . o
, f Fe, CO, ancl Ni are listed in Table 2. 'rhc latter are consitlerably larger ( h i 1
the volume magnetocrystalline anisotropy constants; as a consequence, small strains may
give rise to a n important anisotropy. In particular, this strain-induced anisotropy plays a
very irnportant r6le in ultrathin films, where considerable strains may result, from the epitaxial growth of the film onto a substrate having a different lattice parametcr. A detailled
discussion of this problem is given in Ref.[7].
As a system under strain acquires some magnetic anisotropy, conversely, the existence of a Iion-zero magnetization M along a given direction i - 2 ~induces an anisotropic
st rain of the ferromagnetic body. This particular case of the thermodyrianiic reciprocity
relations is called .magneLostrictaon. The explariatiori of this behavior is quite simple: as
tlic magneto-elastic energy is linear. with respect i,o the strain E , it is always possible for
tlie system to lower its encrgy by acquiring a non-zero strain; this trend is compcicd, of
course, by tlie elastic energy, which is yuudrutzc with respect to the strain. T h e magnitude
ol this spontaneous strain is given by the competition between the elastic and Inagnetorlastic terms, and it depends on the magnetization direction. As the elastic constants are
of the order of
erg.c~ri-~,
i.e. considerably larger that thc magneto-elastic energy, the
rriagnetostriction (relative change of length of the ferromagnetic body) is of the order of
For a cubic system and an hexagonal system, the expressions of the elastic energy
24.9
Tddr 2: Magneto-elastic constants of Fe, CO,and Ni, at room temperature.
~
Fe
U1 (erg.clIi-3)
(eV.alolrl-')
B, (ergcin-")
(eV.aton1-I)
B:j (erg.cui-3)
-
(P\~.atoIll-')
(erg.cm-")
(eV .at om-' )
Ni
CO
(hcc)
-3.44 x i 0 7
-2.53 x10-"
7.62 x107
5.56 ~ 1 0 - 4
-
(hCP)
-8.10 xi07
-5.63 X ~ O - ~
-2.90 X l O S
-2.02 x 1 0 - 3
2.82 X I 0 8
-1.96
-294
-2.05
(fee)
8.87 x10'
6.05 x10-4
1.02 x108
6.97 X ~ O - ~
-
xIO-~
XlOX
-
x10-3
illltl
+
It is tlicri straightforward to minimize the sum G f , i . ( ~ ) C;rnagn,(,l.(S2~. E ) with respect
to E . Oiic ohtains the relative spontaneous magiietostriction (relative change of leagtb)
aloiig thv direction U G (iijl, p2,/ j 3 ) as a function of the iiiagiictization direction
=
( N I , cv2. a:]);for culiic systems, o ~ i egets
i ~ . i i c l1001. hexagonal
systems,
24.10
Table 3 : Magnetostriction constants of Fe. CO, and Ni, at room temperature. (a)Ref. [8];
( b ) R ~ f ."J]; (C)Ref. [lo]
Fe(")
(bcc)
XlOO
All1
CO('))
WP)
= 24 x lo-"
= -23xlo-fi
.-
-
= -5Ox10-"
AB = - 1 O 7 ~ 1 0 - ~
Xc! = 12Gx10-6
AD =-105x10-6
AA
Ni(b)
(fee)
XlOO = -66 x lo-"
Xlll
= -29 x
-
(31)
111(. Iiiagiirtoslrictiori constants of Fc, Co, and Ni are givcn in Table 3. ?'lie rrlsgilc.toelastic constants of Table 2 were obtained from these data [8-10] and from t h e elastic
constants given in Ref.[ll].
r l
3
Anisotropy Arising from Dipolar Interactions
111 t Iic present Section, we aim to discuss, at a microscopic level, tlie rnagiict.ic anisobropy
t l u t ~to tfic dipolar intcractions. In an itinerant rerrormgnet like Fe, CO,or Ni, the ~riagnc%ic.
rriorrierit is riot localized, so that one has to consider the local densit,y of rriagnetizatioIi
m(r) (it should not be confused with the niacroscopic magnetization deiivity M(r ) , which
is averagcd over a large nuiiiber of atomic cells). Thc dipole-dipole interaction has Ixxn
cliscussed by Jansen [12] from the point of view of rclativistic density functional thcory,
which is the appropriate starting point for this problein. The cxpressiori of t h e clipoletl i pole Hamilt oiii an is
where i?i( r) is the inagnctization density operator, expressed in p~ per unit volume. This
result is clearly i~iterpretedas resiilthg from the interaction between the magnekization
and t h e dipolar field created by the magnetization from the whole ferrornagnct. It is a
many-body Harniltonian, which we treat in a I-Iartree approximation, so that tlir dipolar
E,lip. is obtained by replacing in %dip, the operator &(r) by its expectation value m(r).
24.1 1
T h a i the dipole-dipole interaction is a relativistic correction appears clearly, for it is
proportional t o pi
c-~.
11' tlie rriagIietizat,ion distribution within each atomic cell is not spherical, then its
cxparisioll i n rriultipoles includes not only a dipolar moment, hut also higher multjpoles
like qi~acdrupoles,octupolcs, etc. However, in 3d traiisit,ion metals, the magnetization distribution is almost spherical, and can safcly be replaced by the dipolar magnetic- moments
mi (i being the atom index), SO that thc dipolar energy writes
N
(33)
Itcmmbering that d l rnornerits are parallel, as a coIisequeIice of the dominating exchange
interactlion, E,lip. may be rewritten as
(34)
wlierc O,, is the anglc lM,ween G?,I/Iand tlic direction U,, of the pair ( i , j ) ;the latter expresrlcarly displays tllc faart that) dipole-dipole interartion coritrihutes to the magrwtic
anisotropy. For a given pair (z,j)tlie dipolar energy is minimum whcn the moments are
par.?LIIP1 to Ut3.
sioii
3.1
Shape ailisotropy
A st.r.ilting fmturc> of tlic dipolar interaction is t h a t it decreases slowly as a function of
t h e distance T , (like
~
T i ' ) ; thus the suniiriatiori over the pairs ( i , ~converges
)
very slowly.
As a conscqucnce, tlie dipolar field H(lir).(i)experienced by a given moment m, depends
sigtlfi(.illltIy on t I ~ crrioment s located at,' the Ijoundary of t,Iie sample, ancl this resu~tsin
1I l C
ShCll.'r? ( L 7 1 k O ~ T O ~ ) ! ) .
Intuitivcly, wc fcccl that thc contribution to Hcl;,.(z) of atoms that are very far
rroiii i should iiot depend 011 their exact positions at the atomic level, so that one can
safcly rcplacc tlic individual nioinents by the (macrcjscopic) continuous magnetization
distribution M(r); this, however, docs not hold for moments that arc close to i. These
considerattioris are accounted for quantitatively in the Lorentz method for calculating
HcliI,.(i),
which is sketclied in Fig. 3. In ordcr to calculate Hdip.(i), Lorentz decomposes
tlie saniplcl into two p m t s in a spherical cavity of radius R centered a t site z, the discrete
rnorncrit distributiori is retained; in the rest of the sarnpk, the rnoment distribution is
approxiniatcd by tlie macroscopic magnetization density M(r). Of course, the larger R,
the, bcttcr the approximat,ion. For a continuous magnetization distributiori, the dipolar
ficld may he expressed as due t,o psoudo-muynetac charges with a volume density p =
-d i v M and a surface density CJ = 11 . M, where n is the normal to the surface.
If tlir magnetization is uniform, then only the surfaces carry some pseudo-charges.
Thus, Hdip.(z) can be written
where Hcav. is due to the dipoles inside the cavity, HI, = (4n/3)M (the Lorentz field) is
the field crcated by the pseudo-charges at the surface of the cavity, and Hd (the demagnetizing field) is due to the pseudo-charges on the external surface. The sum of the cavity
24.12
-
-
-
I
- -
Figure 3: Sketch of the Lorentz decomposition for calculating the dipolar field Hdilj.(i) experienced by the moment located on atom i .
and Lorentz fields converges rapidely enough (like a sum over ri5) to be estimated with a
moderate cavity radius R; it contributes to the magnetocrystalline anisotropy, and will he
discussed further in Sec. 3.2. The shape anisotropy is entirely due to the demagnetizing
ficld Hd.
Thus, the shape anisotropy is given by
The magnitude M(r) is essentially constant, equal to the bulk value A417 throughout
the sample, and zero outside; however, near the interface, it can deviate from MV (this
deviation accounts for the possible enhancement or reduction of M in the ferromagnet. as
well as for the possible induced magnetization in t h e neighboring material). Thus, we can
separate the total shape anisotropy into a volume term and a surface term. The volume
term is obtaincd by taking M equal to its bulk value, whereas the surface term is due to
the departures from MV near the interface.
For a body of arbitrary shape, the dipolar ficld Hd(r) depends on the position r;
however, if the body has the shape of an ellipsoid, Hd has the striking property of being
uniform (in magnitude and direction) throughout the sample. It is commonly expressed
as
Hd = -4nD * Mv ,
(37)
where D, the demagnetizing tensor, can be shown t o satisfy
Thc sliape anisotropy per unit volumr: then is
24.13
The demagnetizing tensor for simple limit cases may he found easily by symmetry arguments. For a sphere, one has
(;
1/3
D=
0
1f
l;3)
;
for a infinjtc revolutioii cylinder of axis parallel to z ,
D=('!
182
%)
;
and for a plat(: of infinite lateral extension, with the normal parallel to
D=(!
:).
5,
(4%)
hnalyt ical c~xImssionsinay also be ohtained lor ail cllipsoid of revolutioIi. Let (1 is the
polar semi-axis, and b tlic equatorial semi-axis, with m = n / b ; for a prolate ellipsoid
(111 > I ) , one fincls
and for
ail
oblate ellipsoid
(771
< l),
(-14 )
tlic other tensor clcments are obtained by using Eq. (38)) i.e. DI,
= U , = (1 - f l o ) / 2 .
Thc~(as(' or a plate of' infinite lateral extension is relevant for layered systcrns such
as idtrat,liin films and rnultilayers; for such systems, the volume shape anisotropy i s
(46)
and wlicw 13 is t.hc angk between the norrrial to the plane ancl ft,~.It hvors an inplane orientation of i2n.1. For Fe, CO)and Ni, 27~M,$is respectively equal to 1.92 x 10'
erg.cni-3 (= 1.41 x lo-" eV.atom-'), 1.34 x lo7 e r g . ~ r n - ~
(= 9.31 x
eV.atom-'),
and 2.73 x lo6 erg.cm-3 (E 1.18 x
eV.atoin-'). These values are larger than the
volume inagnctocryst,alline anisotropy constants (compare Table 1)) so that, in comparr7I ivcly tllick filrns, t,hc shape anisotropy dominates both the volume and the surface
iriagiietocrystallirle contribut,ions, and the magnetization lies in the film plane.
The surface cont,ribution to the shape anisotropy is easily calculated by considering
infinitesimal sliccs parallel to the surface, and one obtains, per unit area,
24.14
with
in the above equation, 2 < 0 (respectively z > 0) corresponds to the interior (respectively
exterior) of tlie ferromagnetic body, and the excess surface magnetization hi's per unit
a r ~ is
a defiiied by
L
0
AI,? =
[ M ( Z )-
Ill"]d z +
I'"
Ad(*) (12
.
(49)
'I he iriagriitude of the surface sliape anisotropy can be obtained from electronic structure
calculations of the layer-clependerit IriagnetizatioIi near surIaces arid interfaces. For instance. for Fe [13], the magnetization is enhanced a t the surface Fc(001), and onc obtains
I<shape
s
- -0.27 crg.cni-'; the enliancement is slightly less at a Fe/Ag(001) interface, and
one has
= -0.12 erg.cm-2. For Ni [14], one oblairis fjp&a,,e = -0.017 erg.crn-'
for the Ni(001) surface. and
= 0.025 crg.cm-2 for t,he Ni/Ch(001) interface, wlicre
the iriagnrt ization is reduced. Tliesc examples, as cornparecl with t lie orclers of rnagiiitticlr
of A'-' giveri in Sec. 2.1, indicate that, althougli it is riot completely negligible, the shape
5iirface aiiisotropy contributes only weakly to the total surfclcc aniwtropjr In part i r d a r ,
iii any case. thc shape surface anisotropy can 7wiicr lead to a pcrpciid~cularrasy u i s iii
id 1rat h i 11I Ins.
3.2
Dipolar crystalline anisotropy
We consider now t he contxiliution of' the dipolar iiiteractioris to t1w t ~ ~ i ~ ocryst
g r l ~ talliric
aiiisotropy. Its calculation involves nurnerical summation of tlw dipolar ljvltl I'roni the
dipoles locat,cd inside the Lorentz cavity. To do this efFiiciently, sophisticated tcchniques
iriust he usecl. such as tlie Ewald suInmation method, where the surnination is perforrticcl
partly in the real space and partly in the reciprocal space [15].
'1s inay hc seen from Eq. (34), the dipolar energy coritains only terms ol' order 2
witli rcspc>ct to $ 2 ~ T
. l i ~ iit~ ,contributes only to anisotropy constants of orclcr 2. As a
consequence, for structures of high symmetry (such as cubic structures) where tcmx of
ordrr 2 are forbidden, the net dipolar contribution to the iiiagnetocry8tallin(, anisobropy
vaiiishes. Note that, for cubic systems, a non-zero anisotropy would arisc fro111 liiglier
terms i n the riiultipolar rxpansiori of the rnagnetjzatjon density, but this IS quant italively
ricgligible.
On t h e otlicr hand, for structures of lower symmetry, where terms of order 2 are
iillowed, the dipolar crystalline is iii general non zcro. For hcp systems, tlie cli polar
crystallinc anisotropy is foiuirl to he exactly zero for the idral ratio r / u =
M 1.633
ancl t o c~epnrtfrom zrro as c / u departs from
for CO, one ~ i a sc / n = 1 . ~ 2 a2n d the
dipolar contribution to Ii'y is 1
'
: dip. = 5.7 x lo4 e r g . ~ m -(z
~ 4x
eV.atoiri-') [24].
Quaiitit atively, this contributioii to the volurric anisotropy of hcp CO is iiegiigihle.
As was pointed out in Sec. 2.5, the symmetry of cubic crystal under strain is
lowcretl. so that anisotropy t e r m of order 2 become allowed. Thus, tlic dipolar interaction
24.15
Table 4: Chlculated dipolar contribution to the magneto-elastic constaiits of Fe and Ni; from
lief. [I 71.
Table 5: Calcnlat8edvalues of the dimensionlefis parameter ks characterizing the inagnitudc of
the dipolar contribution to the inagnctocrystalline surface anisotropy. for various systems; from
Ref. [IS]
0.03
ICS
- 0.034
-0.118
-0.038
-0.218
-0.0:34
slio~iltlcoritribui~rto the magneto-clashic colistsaritsBl and B2 of cubic materials. Explicit
calculations give 1171
(50)
with 11 z 0.8 aiid 0.6 for the 1xc and f'cc structures, respectively. The corresponding
r c w ~ l t sfor Fe and Ni are given in Table 4. Again, these values are considerably smaller
h a n tlic cxperiirierital OIICS, so that otlicr contrihutiotls should dominate.
As eIrip1iasizcd i n Scc. 2.4, the local syrrinietry is lowcred a t a surface, so that, even
lor cubic crystals, tlie dipolar interactions give a tion-zero contribution to the surface
cryst,allinc anisotropy. As the Lorentz inethod cannot be used for atoms located near a
surface. one needs another type of' decomposition in this case [18];the expressioll of the
dipolar contribution t,o 1'; is
(51 )
wliere d is tlie tlistarice between atomic planes and ks is given in Table 5 for various
surfaccs. The largcst value is obtained for the Fc(O01) surface: K f dip. = 0.06 crgcrn-'
( E 0: x
eV.atoni-'); again the dipolar interactions yield only a very small contribution
to t lir ol~sewcdsurface crystalline anisotropies.
1o suinriiarize t hc prcsent Section on the magnetic anisotropy due to dipolar interiLctions, wc have examined i n detail the contributions of the latter to the various terms
of th e total anisotropy energy B(i?,kI). In all cases, we have given a quantitative estimate
of tlic dipolar coiltribu(,ion. It turns out that the essential contribution of the dipolar
iriteractions to I ; ( s l ~is) the volume shape anisotropy G'shape(nfif). For all other terms
(magneto-elastic anisotropy, volume and surface crystalliiie anisotropies), the dipolar contribution is quantitatively not important, and, as will be shown in the next Section, they
must be attributed esseiitially to the spin-orbit coupling.
I
,
24.16
4
Anisotropy Arising from the Spin-Orbit Coupling
In this Section. we shall proceed in steps of increasing sophistication. After having presented the spin-orbit interaction, we propose a simple model for the magnetic anisotropy;
then. we present the perturbation theory, and finally, we discuss the state-of-the-art of
first-principles calculations of magnetic anisotropy. The discussion will be focussed on
a few selected examples, and we shall not attempt to mention all the works that, have
been published on this topic. Otherwise explicitly specified, in all this Section, the term
niagnrtic anisotrop!/ will refer to the contribution arising from the spin-orbit coupling.
4.1
The spin-orbit coupling
The relativistic theory of the electron relies on the Dirac equation. In the limit of low
velocities (more precisely to order v2/c2), the Dirac equation reduces to the l’auli equation. which is essentially a Schrodinger eqiiatioii with relativistic corrections; the Pauli
Hamiltonian writes
The interpretation of the various ternis is as follows: The first two ternis are respectively
the non-relativistic kinetic energy and the electrostatic potential eiiprgy; they form thc
non-relativistic Harniltonian. The third term is the relativistic mass-velocity correctioli.
7 he fourth term is the Darwin correction, which accounts for the fact that, within Lhe
relativistic theory, the electron is sensitive to the electric field E over a lengtliscale of
the order of the Compton wavelength Xc = h/(n7c). The third and fourtli t e r m are
independent of the spin s = u/2;they are often combined with the non-relativistic terms
to form the so-called scalar-relatzvastzc Hamiltonian. The last term in Eq. (52) is tlic
spin-orbit coupling ‘Idso . It can be interpretcc1 as the coupling hctween tlic spin of tlie
electron and the magnetic field created by its own orbital rnotiori around tlie nucleus. As
the orbital motion itself is directly coupled to the lattice via the electric potcntial of tlie
ions. this term provides a contribution to the magnetocrystalline anisot,ropy.
The spin-orbit term is large essentially in the neighborhood of tlie nucleus, where,
t o a fairly good approximation, the potential is spherically symmetric; tlieii the electric
field writes
(53)
so that tlie spin-orbit Hamiltonian may be expressed as
As the riiagnetisnl of transition metals is due to the d electzons, it is sufficient to consider
only the spin-orbit interaction for d electrons. Thus, the spin-orbit coupling finally writes
l-ts.0. = t 1 . s
7
(55)
where E , the spin-orbit constant, is the radial average of [ ( r ) over d-orbitals. The c d culated spin-orbit constants of transition metals are shown in Fig. 4. It appears that
24.17
Table
(j:
C~yroInagocticfart80r'g and orbjtal moment nzr = //,BIZof Fe, CO, and Ni; from Ref.[20]
Fe
2.091
0.0918
9
1111
(pB.atoni-I)
CO
2.187
0.14'72
Ni
2.183
0.0507
iiicrcascs considerably with increasing atomic number 2 ; within a given series of the
periodic table, it increases like 2'. For the 3d metals, E is of the order of 50-100 meV.
In a free atom or ion, the Hamiltoiiian is spherically symmetric, and the total orbital
irioineiit L a good quantum number; thus, the ground state has a non-zero orbital moment,
accordiiig t o llir 2Iid Hund's rule. This is essentialIy the case for rare earth 4s ions.
On thc othrr hand, for 3d transition ions, thc clectric field of thc neighboring ions
(the c y p t a l held) healcs the spherical symmetry. 'l'hc energy of the crystal field is typically of tllc order o l 1 eV, i.e. large as cornpared to the spin-orbit coupling, which can
d iri a first approximation. Bccause of thc crystal field, the energy lcvels no
loiigcr corrrsporicl to a dcfiiiite quantum number 7111; rather they correspond may be Iabelled as z y , yz,zx, x 2 - y2, or :3z2 - r', which are hybrids of opposit orbital rrioirierlt
~ 1 and
1
--1n/, so tliat the rict orbital rnoincrit of these levels is zero: the orbital moment
i s said to lie grit nrhrd by thc crystal field. 'I'hus, in absence or spin-orbit coupling, the
~tiagiicxticr r i o r n c n l of 3 d ions would be purely a spin moment, and the gyromagnetic factor
.r/ = (2.5,
LZ)/(.sz lZ) would l x equal to 2. The effect of the spin-orbit coupling is to
r ~ ~ 1 1 i o111
w part t h yucricliing of tlie orbital moment, but t lie efFect is rather small, and
tlic gyrornagnctic factor y rernairis close 10 2.
'I'lic> s i m o clrcct Iiappens in 3d metals, wticre the rble of the crystal field is played by
t,hc 1)antl dispcrsion of thc Icvcls; inderd hi Fe, C h , and Ni, the orbital moment is almost
qucnclicd arid the gyroniagIietic factor is rlose lo 2 as can he seen from Table 6.
We lisp for tlie spin components the rotated frame (Oc,O S ,( I ( ) . The matrix eleiwnis of 1 . s d ~ p c i i don the quantization axis O( of's, which wc naturally chosc along the
rriag~ic~tizatioii
clirciction f 2 ~ Strictly
.
speaking, the arigular mornenturn and its magnetic
inoiricmi arc ant ipillall(J1;for simplification, wr talte them to bc parallel, or cquivalcntly,
we ~ a l w/ L H iiegative. Tlie olwrator 1 . s then writes
-+
+
(56)
wlicw 1$ = 1, f l,,. It t h r i a sirnplc rnai,t,er tlo calculate explicitly the matrix elernenis of
1 . s between the various d orbitals (zy, x2 - y2, etc.) as function of the angles 8 and 4;
tlic rrsiilt, may bc cxprcssccl as
(57)
whcw t lie 5 x 5 matrices M arid N are givcn in Table 7.
4.2
Siiiiple physical picture
Before giving a detaillcd discussion of the microscopic theory of magnetic anisotropy, we
first proposc a simple physical picture. Let us consider the case where the exchange
-
24.18
Z'
so00
6000
I
5(
4c
-
&Y
a
E
v
U
20
?
Rh
10
I
I
I
Z
'
Figure 4: Calculated values of the spin-orbit constant [ of transition metals; from Itef.[lS]
24.19
TabEe 7: Matrix elements of' M a.nd N : the labels 1 to 5 correspond respectively to the q ,
yz,
22, :c2 - y 2 , and 3z2 - r 2 d-orbitals; from Ref. [21].
I1 I >
A/,
<Ill
<2 1 I
I
<:31
<4 1 I
<6
I I
N
<I11
0
-
12
+ sin
i
i isinesing
13 I >
I>
8 sin cb
-
0
i cos e
+icos~
- iisinHcos4
$isinflcosgh
-$icose
0
-~COSO
iisinBcos4
iisinOsin4
0
I1 I.>
0
4& i
14 I >
3 i sin H co6 4
-
,i sin Osiri (I,
3
(CO5
I>
(sin 4
13
#l
ti cos 6, sin 4)
+
-i cos 8 cos 4 )
-
+
sine
- ifiisiriBcos4
&i
0
14
sin6sin 4
0
0
sin B cos 4 - & i s i n 6, sin 6
12 .1>
15 'I>
0
0
I>
15 .I>
- i sin 0
0
4 (sin 4
&(sill 4
- i cos 0 ('OSd )
-i cos ecos 4)
4
0
wit 11
(59)
For a system of hexagonal or tetragonal symrnet8ry,with thc symmetry axis along Oz, the
orbital susceptibility writes
Thus. tlir orbital contribution to thc rnagnctic morncnt writcs
arid tlic spin-orbit, energy is
(65)
or
c u l i c syrnrnetry. tlie only itic~ependrntelenient,s of
that the expressioiis ol
/,L!
x'')
orb.
are
(4)1111
orb.
and Es.0. for cubic systems arc, respectively,
.
ancl y (4)1122 so
0rh.
Siniilarly, liiglier order anisotropy terms arc taken iiito account by considering higher
orclcr t c r m in the non-linear susceptihlity.
Although tlic limit of large exchange coupling does not hold for Fe, CO and Ni, wc
xiay tentatively apply this model to the latter. For these metals, the efrective field Horl,
is of the order of 5 x 10' Oe. The orbital susceptibility of Fe, CO and Ni is given in T&I&
8. Data on the non-linear susceptibility are not available.
24.21
'lhble 8: Orbital susceptibility of Fe, CO,and Ni; (")Ref. [a]; (b)Ref. [5]
Fe
CO
Ni
Tlie predictions of the present inodel are the fallowing: (i) the orbital iiioineiit is of
thr order of 0.1 p~.at,oin-', parallel t o the spin moment (i.e. y > 2); (ii) K 1 arid &I1 in
Table I arc ol opposit sign, a i d the ratio -2K1/MI is of the order of Horb ; (iii) for CO,
tlie rnoinetit, anisotropy M , is of t81ieorder of 10-2 pB.atoni-1. AII these piwlictioiis are
fairly wcll salisficd, both in sign antl order of magnitude, which indicate tliat the siniplc
pliysiral picturr proposed hrrc is csseiitially corrcxt,
4.3
Perturbation theory
Siiicc 1 hc spin-orht coupling ( is much smaller than the handwidth aiid the exchangc
splitting, it8 is quitc natiiral to attack tlie problem of calculating the risagnrtic anisotropy
I)y iising t hc prrt,url)ation theory.
As inay bc scm from 'l'able 7, tlie matrix elenleiits of 1 . s are comhiiiatioiis of first
order spherical harmonics of f 2 ~thus
; the matrix elements of (1 . s)" are combinations of
splic~icnlliarrrioriicu ol order ri. So, in ortlcr t I) calculate im anisotropy constant of order
? ) , o r i c ~has to use' pc~rtiirbat,iontlwory of ordcr 71. For licp crystals arid ultratliin films,
P('o r ( ~ eI)('ttiirl)cition
r
is sullir*ient,whereas for c*ut)i( crystals rrihgiietic anisotmpy arises
oiily in 4''1 ortlrr pvrt tirI)a(ion theory.
'rlie cliange in enc'rgy to Z1ld ordw in spin-orbit coupling is given I)y the well-known
(69)
wlic~c.Ilie Ial)rls "gr." arid "cxc." refer t,o the unperturbed ground state antl excited
s(iit,chs, rrsprctivc>ly. Tlie only cxcitctl states orir nceds to consider here are those where
an c~lcctroiiof momcmtum k is raised from a n occupied state into an empty state above
the Fermi Icvcl, with or without spin flip.
'I'lius, a vcry rough rstimatc of Ir'l for a uniaxial syst,em, is
K,
- t2 '
-
I4f
wIicre W is tJie d I)andwidt,li; similarly, onc may estimate t h e anisotropy of cubic crystals
frorrl $111 ortlw pt'rturl)at ioti t,lieory to 11e
lil
N
E*
w3.
(71)
Taking E M 75 ineV and Mf M 5 eV, oiie obtains li'l w 1 meV.atotn-' for a uniaxial system,
aiid K1 M 0.3 peV.at,om-' for a cubic system; these rough estimates are respectively
24.22
of the ordcr of inagriitutle of the observed anisotropy in ultrathin films and bulk cubic
ferromagnets, respectively. Thus, the perturbation theory provides a sirnple exp1anat)ion
for the order of rnagriitude of the magnetic anisotropy. Quite generally, we conclude that
anisotropy constants of order 72 are given by powers of order 72 of ( / W which is a sinall
quantity: this explains the rapid convergence of the expansion of the anisotropy energy
in spherical harmonics.
I n the following, we restrict ourselves to uniaxial systems, i.e. to Znd order perturhation throry. By using the symmetries of the matrix elernents of 1 s, one gets
LET() = - t i
< 7r11 1 11 - slmz T><mCjt 11 . slm4 t> c(7nl,m 2 ,
wig, m4) I
n i l ,1n~,ni3.mp
(72)
where G(nil. )r22r 1723, m4 ) depends only on the non-perturbed band structure, while the
matrix elenients of 1. s depend only or1 C2bf. More details may be found in Ref. [22].
Finally. one' obtains
1Eso. = I<,, t lil sin2'6
(73)
wliere Ii'l is proportional to t 2 .The virtue of the pertulbation theory is that is allows to
calculate d i r c d y t hc anisotropy constants without calculating explicitly tlic tobal erwrgy
of the systcm as a function of t Iic dirc3ctioii of thc rnagnctization. 011 t h oLliu liarid,
it has the inconvenient of incorrectly handling degenerate levels and dcforrrialioris or thc
Fermi surface.
We give liere an exrmple of calculated magnetic anisotropics for thr rase of fcr
(001) and (111) monolayers [XI. The band-structure has been calculated by using the
tight-binding method, incliiding 3d and 4s bands. Special attention tnusi, I x paid to
the low convergence of the integration over the two-dimensional 13rillouin zoiic: in the
present casc it was necessary to perform the siimniation over more than 5000 k-points.
The results are shown in Fig. 5 . In order to investigate the trends accross tlie :Jd scricu,
111
' has bccn talculatcd as a function of tlie number NV of valelice c4ec!rons, for Nv = 7
(hypothetical ferromagnetic Mn) to NV = 10 (Ni). The most striking trend (indicated by
the dashed line in Fig. 5) is a sytematic variation with respect to N v : for Nv < 8 tlie
anisotropy favors a perpendicular Inagnetizatioii, whilc for NV > 8 is favors i~11 i ~ ~ - p l a n r
riiagricti~atioii.This trend has been intcrprctcd in connection with a thwrcrn stating tllat
f i l must change of sign at least 4 times as the number of d electrons increases from 0 to
10 "231.
It IS interesting to note that the sign of the fcc (111) COmonolayer is in contradiction
with the perpendicular magnetization observed in ultrathin fcc (111) or hcp (0001) CO
hlrris; this point will be discussed further in the next Section.
4.4
First-principles calculations
The ab t n i f z o calculation of the magnetic anisotropy is a Iormidablc task. Tlir usual
procedure is to compute the difference of total energy for C2211.r along two non-equivalent
directions (for cxample [001] and [111] in a cubic crystal). In a (bulk) cubic crystal, the
anisotropy energy is of the order of
eV.atorn-', while the total energy per atom
is about 40x103 eV.atoin-I; thus the total erieigy would have to Le calrulated with a
trcnieridous nunicrical accuracy. Such brute-force calculations are of course not feasil>le.
Fortunately, most contributions to the total energy remain (almost!) unchanged upon
rotation; thus they can be skipped in the total energy. The frozen-core approxirnatiorl
24.23
-1 .'
-2..
Figure 5: Calculated anisotropy constant K 1 of transition metals fcc (001) and (111) monolayyers,
a function of the number NV of valence electrons; the dashed line i s a guide for the eyes.
From lhf. [22]
a,s
and tlic use of thc force thcorcm allow t o obtain the total energy difference as the c!ifFercnce
between tlie sums o f one-electron energies; the latter are of the order of 10 eV.atom-I, so
that the calculation of the energy clifierence with an accuracy of 10-6 eV.atorn-' remains
extremely difficult. On the other hand, the magnetic anisotropy of interfaces is in th c
range of lo-' to lo-' eV.atoin-l, and should be calculated much more reliably.
Thus, the usual procedure for calculating the magnetic anisotropy from first principles is: (i) perform a self-consistent spin-polarized calculation for the scalar relativistic
I-Iamiltonian; (ii) perform a (non self-consistent) calculation including the spin-orbit coupling for various directions f 2 and
~ take the difference between the sums of one-electron
eigenvalucs. The band structure is generally calculated within the local spin-density approximation (LSDA), and the Schroedinger equation is solved by using a linear scheme
such as the linear muffin-tin orbital (LMTO) method, or the linearized full-potential augmented planP wave (FLAPW) method.
For bulk transition mctals, the most detailled calculations arc due to Daaldcrop e1
al. [24], who used the LMTO method. In spite of extremely careful calculations (they used
up to 500,000 k-points for the Brillouin zone integration), they did not obtain results in
agrcemcnt wit11 experimental data: for Ni and CO, they predict a wrong easy axis, while
for Fe, they obtain the correct sign for the anisotropy, but a factor 3 too small. The
order of magnitude is nevertheless correct. They have investigated the possible origin
of the discrepancy between the calcuIated and experimental results, and suggested that
the latter might bc due to incorrect positioning of some degenerate bands near the Fermi
surfacc.
As alrcady outlined, the situation is much favorable in the case ultratliin films,
wherc tlie anisotropy is scvcral ordcrs of magnitude larger than in bulk materials. Wc
24.24
Table 9: Calculated anisotropy constants K l of fcc CO (111) monolayers on various substrates;
m ( S ) is the induced magnetic moment in the substratc at tlie interface.
system
method
K,
(meV.atom-l j
vacuuni/Co/vacuum
vacuurn/Co/vacuum
LMTO
FLAPW
c 11/ C O / c'u
LMTO
Ag/C'o/Ag
I'd / ( '01P cl
vacuurn/Co/Pt
LMTO
LMTO
FLAPW
4s)
Ref.
( p .atom-')
~
- 1.20
__
1251
-0.65
0.20
0.18
0.82
0.45
-
[26]
< 0.01
< 0.01
0.30
0.37
~ 7 1
1271
~ 7 1
ps]
shall discuss here the case of fcc (111) (or hcp(0001j) CO ultrathin films, which have
bc~rnwidely irivestigated c~xperinieritally,and cxhibi t a perpendicular magnetization wlic~n
sandwiched between Au. Pd, or Pt.
The results of tlic calculations are presented in Table 9. One Iirst rrmarlcs that the
anisotropy of tlie free-slanding CO ( 111j monolayer strongly favors an in-plane orientation
of the magnetization, in agreement with the results of perturbation theory. 'I'liis is in
contrast with the case where the CO layer is in contact with a substrate (CAI, Ag, I'd, 1%):
the anisotropy now favors a perpendicular magnetizat,ion; this tendency is particularly
strong for Pd and Pt (note that for the latter, the anisotropy was calculated with only
one (.'o/Pt interface, so that an even larger anisotropy may be expected for a Pt/Co/Pt
sandwich). These results are essentially in agreement with experimeiil.
111order to understand the r6le played by the substrate in establisliirig tlic pc!rpentlicular anisotropy, it is important to note that both Pd and P t have a large Stoner-enhanced
susceptiblitp together with a large spin-orbit coupling. Thus, they acquire a sizeable spinpolarization at the contact of CO and give an important contribution to the anisotropy,
due to their large spin-orbit coupling. This interpretation is supported by the fact that
suppressing the spin-orbit interaction in Pd strongly reduces the calculated anisotropy of
Pd/Co/Pd films [27].
On the other hand, C h arid Ag have filled d bands, and the induced sI'iIi-polarization
is nrgligible; thus, they infhence the anisotropy only via the s-d hybridization with tlic
C'o d bands. and their spin-orbit interaction does not play an important r6le.
Another very interesting case is that of fcc (111) Co/Ni multilayers, wliere both
constituents are ferromagnetic mctals. Daalderop et al. noted tliat since Ni is isoelc!ctroIlic
of' Pd. a perpendicular anisotropy could be expected for this sytem as well: from LMTO
calculations. they predicted a perpendicular easy axis for Col /Ni2 Inultilayers, a result
which they confirmed experimentally [ B ] .The latter result is a brilliarlt success of the
theory of magnetic anisotropy.
5
Outlook and Conclusions
I n this Chapter, we have attempted to make clear the mechanisms by which small relativistic corrections to the Hamiltonian (the dipole-dipole interaction and the spin-orbit
coupling) give rise to the important property of magnetic anisotropy. The order of magnitude of the various contributions have been estimated and compared with experimental
24.25
data.
Thc>eflcct of' the dipolar interactions appear to be almost entirely contained in thc
shape anisolropy, which (leperids on the magnetic material in a rather trivial way: via the
magnitude of the (bulk) magnetization Mv,
The anisotropy due to the spin-orbit coupling, on the other hand is very subtle, and
depends in a complicated manner on the band structure of' the material, which makes its
calculation a very difficult task.
Ncvdiclcss, by using simple arguments, we can explain thc rclationships between
the crystalline symmetry of the system and the magnitude of its magnetic anisotropy:
thus, we have a satisfying explanation for thc fact, tliat low-symmetry systcms, like ultratliiii films, exhibit mucli larger anisotropics than bulk cubic materials. With the Iiclp of
a simple model, we have been able to explain the conneclion between the anisotropy of
t h c cnergy and those of the magnetization and orbital susccptibjlity.
'I'lie st atr-of-the-art of hit-principles calculations of magnetic anisotropy lor bull:
materials is rathcr disappointing: more than half a century after t h e pionneering work
ot I3rooks [ X I ] , we are riot able to explain from lirst-principles why Fe, CO, and Ni are
respectively magnetized along the (100). (WOl), arid [ 111) axes! 'Yhe situation loolcs
hetter, howcver, for the casc of low-symmetry s y s t e m whrrc very encouraging S I I C C ( ~ S S
has berii obtained, i n particular for ultrathin films.
24.26
Textbooks and review articles on magnetic anisotropy that have been used for
the preparation of these lectures
o
R.R. Birss, Symmetry and Magnetism, Selected To1)ics in,Solid State Physics Vol. 111,
edited by E.P. Wohlhrth, North-Holland, Amsterdam (1964)
P. Bruno, Anisotropie Magnktique et Hyslkre'sis du Cobalt d I'Echelle du P1a.n Atornique,
P1i.D. Thesis, Orsay (1989), unpublished
a
P. Bruno and J.P. Renard, Magnetic Surface Anisotropy of Transition Metal liltlothin
Films. Appl. Phys. A 49, 499 (1989)
o
W.J. (larr Jr., Secondary Eflects in Ferromagnetism, in Handbuch $er Physik, Band
XVIII/2: Ferromagnetismtis, p. 274, edited by H.P.d. Wijn, Springer-Verlag, Berlin ( 1966)
o
S. Chikazumi, Physics of Magnetism, John Wiley & Sons, New-York (1964)
o
G.II.0. Daalderop, Magnetic Anisotropy from First Principles, Ph.11. Thesis, Eindhoven
(1901). unpublished
11.
Ilerpin, T/i&rie
hi
Magne'tisme, Presses Univrrstitaires dc France, Paris ( 1068)
.J. Kaiianiori, Anisotropy iind hfagnetostriction of Frrromagnetic. U 7 l d Anfif~rrornngn~tic.
.\lofrr.io/s. in Magnetism, Vol. 1, p. 127, edited by G.T. Rado aad 11. Suhl, Acadernic
Press. New-York ( 1963)
E;. Kncllcr. Ferrornagiietisiniis,Springer-Verlag, Berlin ( 1962)
L.D. Landau and E M . Lifshitz, fYecfrodyriamicsof
&?7ti?lU071S
Media, Prrga,nion, Oxford
(1960)
A.H. Morrish, The Physical Prinriplrs
(196.5)
0)'
filognelism, J o h n Wilry b% Sons, New-York
W. Nolting, Qunnknfhcorip dcs Muyndisnzus, 'I'eubner, St.ut tgart ( 1986)
24.28
NOTIZEN
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