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New mechanisms for magnetic anisotropy in localised S-state moment materials
Journal of Magnetism and Magnetic Materials I (1976) 214-225 ~) North-Holland PublishingCompany
NEW MECHANISMS FOR MAGNETIC ANISOTROPY IN LOCALISED S-STATE MOMENT MATERIA,I.~ D.A. SM~H Physics Department, Monash University, Clayton, Victoria 3168, Australia
Anisotropie versions of the familiar (RKKY) indirect exchange betweei) localised S-state magnetic moments in metals can arise in a number of ways. There are two basic mechanisms, viz. (i) from anisotropic s-d interactions such as dipolar coupling and (ii) in higher orders of perturbation theory with the conduction electron spin-orbit interaction. Applications to particular materials are briefly considered. 1. Introduction This paper is primarily concerned with local magnetic anisotropy, such as anisotropic exchange or single ion amsotropy, in metallic local moment materials. These are metallic elements, alloys or compounds for which the origin of magnetism resides in localised magnetic moments, which will be restricted herein to "spin only" me. ments, i.e, with g near 2. Examples are Gd, the Heusler alloys, Au2Mn, Znl3Mn [ 1] and indeed all dilute inter'. metallic compounds and alloys of the 3d transition elements provided that good moments exist and]or the tern. perature range of interest is well above the Kondo temperature. The microscopic origin of bulk magnetocrystalline anisotropy in the ferromagnetic transition metals is an old problem of band theory which has been discussed by Brooks [2] and Fletcher [3]. Here anisotropy arises frcm the corabined effects of 3d spin-orbit interaction and spin-split energy bands. This kind of mechanism is not likely to wo:k for local-moment metals becat:se the spin-orbit coupling cannot act directly on an orbitally non-degenerate moment and the conduction band will be only weakly spin-split. On the other hand, the conventional mechanism for single 3d ion anisotropy in insulators [4] requires orbitally degenerate excited crystal field states of the ion which are mixed into the crystal field ground state by spin-orbit coupling. A similar mechanism works even for S-state ions if higher terms (possibly of different configurations) which are themselves crystal field split are mixed intc, the S-state by magnetic interactions [5]. But for 3d moments in metals our present understanding, at least in the opinion of this author, is not yet at the stage where the lack of orbital degeneracy (as evidenced by the g factor) can confidently be ascribed to quenching by crystalline fields, so that any discussion of anisotropy arising from ex. cited crystal field states would be purely speculative. For the purposes of this paper we shall leave this question open by regarding all local moments as S-state ions, thus implying that no orbital moment is required at any stage of the calculations. The s-d exchange model, which treats the 3d ion as a localised spin, is sufficient for most of the processes conside,red here, which are best specified in diagrammatic form. Fig. ! lists the Feynmann diagrams for anisotropic indirect exchange between two localised spins S 1 and S 2 via the conduction electrons. In looking for an anis,gtropic interaction of the simplest firm = -- S'1 • X" $2,
(I)
(where K must be other than a +multiple of the unit tensor) a necessary condition is that there exists a virtual + process giving the interaction S I S 2 whicll changes the quantum numbers of both spins by + 1. In all diagrams an electron-hole pair is emitted from one spin and absorbed by the other. The dominant process, which gives rise to the usual RKKY exchange [6,7], has an s.d exchange interaction of the form S.,~ (s being a conduction electron
D.A. Smith~New mechanism for magnetic anisotropy
M¢1
M1
T
MZ
M¢I
~t't
M1
(a) "~,/"X""P"~
M2
M 2,.1 (b)
I
M~I
~ ,.
MI
V
(c)
Us
215
~
M2
M2¢'1
(d)
m2 (e)
(f)
Fig. I. Feynmann diagrams considered for anisotropic spin cuuplings. Full lines indicate conduction electron propagators and the dotted lines are included to indicate the initial and final states of the two Iocalised spins. • indicates an s-d exchange vertex (10), n the dipolar interaction (1 ! ), O the orbital interaction (i 2) and X indicates spin-orbit coupling (I 3) at some other site. Some diagrams have been spin-labelled to indicate anisotropic exchange, etc. as discussed in the text.
spin) at each vertex: this particular vertex conserves total spin and so gives indirect interactions S~ S~, S~S~ etc. consistent with the isotropic form S t • S 2. The simplest way to obtain anisotropic exchange is to introduce a vertex which does not conserve spin such as the dipolar spin-spin interaction; fig. l a is spin-labelled for an S~s + vertex. This mechanism has been given for the analagous case of nuclear spins by Bloembergen and Rowland [8]. Alternatively, if spin-orbit centres are admitted which scatter conduction electrons then Bloch states of opposite spin are admixed and an S+s" vertex is effectively converted into S+s z (fig. l b). This process also yields single-ion anisotropy if the two spins are replaced by just one, and has already been given by Smith and Haberkern [9]. Both processes yield a uniaxial anisotropy tensor of the form K = K(R) R/~, (2) in which the axis in question is the direction of the vector R joining the two spins or, in the case of single-ion anisotropy, the vector from the spin to the spin-orbit centre. Thus pair anisotropy is not directly re:ated to the crystal axes but to the directions defined by pairs of magnetic ions. The distinction would of course be spurious if this anisotropy were limited to nearest neighbours, but this is not the case. For both mechanisms the asymptotic form for large R is K ( R ) ~-
B cos(2kFR + 9)
,
(3)
(2kFR)3 where k v is the fermi radius, which is long-ranged and oscillatory with tile same power law and wavelength (though not necessarily the same phase) as the RKKY interaction itself. Some consequences of this result arc ex_l___a ltt _, ,Eric: L . ~tlu -_~ o f , Ltttt; - papt:i. ptult:u
The magnitude of B can vary quite widely. For the process of fig. l a it is most easily s~ecifie:l by saying that the coefficient of R - 3 i.e. B/(2kI:) 3, is smaller than the corresponding coefficient (2/~B) '~ of the direct dip,;le. dipole interaction by a factor of the order of J2P, where ,It is the lth partial wave component of s-d exchange and p is the electronic density of states at the fermi level per spin per atom. This factor, which is typically of the order of 0.07 for 3d moments, is exactly the expected ratio of the indirect interaction to the direct one. For fig. l b the value of B is of the order of JI J 2 w2p3 where w is the strength of the spin.orbit coupling, assumed to be centred at the magnetic ions. Hence this process will dominate the direct dipole interactioa for large w; ?. more preclise estimate for Mn in Cu (for which J 2 = - 0.5 eV, p -- 0.15 eV -I and JI possibly = - 0.05 eV) is that w > 0.5 eV. In
21 b
D..4. Smith/New mechanism for magnetic anisotropy
fact the appropriate value of w in QtMn is 0. i 2 eV from spin scattering off Mn sites, from e.s.r, measuremertts by Schultz, Shanabarger and Ptatzman [ 10], so that rather heavy spin-orbit scattering is required. Of the remaining diagrams in fig. !, (c) is an example of a process which cannot give an anisotropic interaction + + at all as it cannot be spin labelled to yield an S 1 S 2 term. (d) involves the orbital magnetic interaction (12) and spin-.~rbit interaction, which may be symbolised by S. I and I. s respectively; this diagram will give anisotropic exdlange as in (!) but its magnitude witl be smaller than the contribution from (a) by a factor of wp. (e) involves the combination 5! • s I.s 52 "s which, on tracing out the conduction spin uroduct, gives Dzyaloshinskii and Me,riya exchange [ 11 ] of the form D. (51 X 52). For D to be non-zero the distribution of spin-orbit centres around the two spin sites must be of low symmetry. In particular, when the spin-orbit centres are at the spin sites themselves D will be zero unless the spin-orbit coupling constants (or s-d exchange constants) at the two sites are different. Graph (~ is similar but will give a smaller interaction because of the dipolar vertex. l'he .same processes can in principle also give rise to single-ion anisotropy, which can formally be extracted by treating the two spins as just one. Then anisotropy can be created only by spimorbit centres at different sites. Thins has already been considered elsewhere [9] but the results will be summarised for completeness. Graph (e) does not contribute, but the dominant contribution is from (b), giving a short-rang,: uniaxial anisotropy of the form
~'
=
-
S.
(4)
F-S.
if there is only one spin-orbit centre close to the magnetic ion then F= F(R ) [~/t,
(5)
where R is the vector between the two sites. For two centres at positions R 1 and R, from the magnetic ion additional tensor forms, not considered in ref. [9], arise and the total contribution looks like F = "r:'l Rlkl + F 2 / ~ 2 R 2 + G(~'lk2 +/~2/~1) + G ' / ~ 1 X k2 1~1X/~2 ,
(6)
where the coefficients are functions ofR ! and R 2. The general case of n neighbouring centres would be described bv a superposition of pairs. There is :ao long-range effect here analagous to the R K K Y interaction; thus F(R) has an asymptotic form proportional to R-7 so that the net anisotropy tensor will be dominated by nearest neighbour contributions.
2. The calculation of anisotropic interactions 2. 1. ~ e ' Hamiltonian and met,hod oj' calculation Consider two S-state ions of the 2d series located at positions R 1 and R 2 in a free-electron-like metal. The ions will be represented by localised spin:; 1~1 and S 2 which interact with the conduction electrons by various "s.d" int, factions, viz. exchange, dipolar s?iLn-spin and magnetic spin-orbital interactions. The conduction electrons also suf :er spin-orbit scattering at sites laaelled by Ra, which will be specified further in due course. Thus the cornpie, e Hamlltonian is it = H 0 + H',+ W,
(7)
wh,:re
Hc, =
+
ko
q.
Cko,
(8)
is the con Juction electron tern: in second quantised form, /f' - *'1:'~ + Hdir. + Hc,rb"
(9)
D.A. Smith/New mechanism.formagneticanisotropy
217
is the sum of the s-d interactions, and W is the spin-orbit interaction. It is sufficient to specify H ' and W by their one-electron forms. Thus for a spin S at the origin of coordinates we write
(klHexlk),
=
~ (2/+ - N 1 t=0,1,2
1)Jl(k,k')Pl(k.k' ) S.s.•
Hdip(r ) "- - (2/.tB) 2 S- (V V 1 ) ' s Horb(r ) ,,. (2#B)2
S'I
r3
(!0)
(r large),
( ! I)
(r large),
(12)
where #B is the Bohr magneton and s and I are the spin and orbital angular momenta of the electron at r. ( 11 ) and (12) are the classical forms valid when the radius r exceeds that of the 3d shell; the exact expressions, which depend on the internal atomic structure of the ion, will not be required. Of the three exchange coefficients, "/2 will dominate as it contains the covalent admixing term given by Schrieffer and Wolff [12]. The one-electron matrix element for a spin-orbit centre at the origin is taken as (klWlk')
-
iw(k,k')
k X k'
N
k2F
"s,
(13)
which is a simplified version of Animalu's model potential [l 3]. The precise dependence of these matrix elements on k and k' is not crucial since it is the angular dependence which determines the form of indirect interactions; thus J/(k, k' ) and w(k, k' ~1will be replaced with their fermi surface values. Of the various ways of deriving indirect interactions, perhaps the simplest is in terms of linear response theory. which will give the energy tetra linear in S 1 and S 2 [7]. The advantage of this method is that it leads directly to an expression in terms of propagators, thus providing an automatic prescription for dealing with vanishing energy denominators. To formalise the method, write H I=-S
1.hI,
H 2=-S
2 . h 2,
(~'+~
where h I and h 2 are the respective conduction electron operators. H~ may be thought of as the static limit of a time-dependent perturbation; this perturbation induces a mean internal field (h2) = ((h2 JH'l ))0 + where ((A [B))z is the analytic continuation of the Fourier coefficient of the temperature Green function of A and B [14]. Here 0 + indicates zero frequency with a positive infinitesimal imaginary part, indicating the retarded response. By considering the torque that this field exerts on the second spin, the energy of interaction is found to be (]5)
~ = _ $ 2 . ( h 2 ) = ((H~ IH'1 )>0+" Moreover, a completely analogous argument treating 11'2 as a perturbation will lead to the same result with H ! interchanged. A hermicity relation for Green functions can now be used to write
H 2 and
((H 2 IU'1))0+ = ((H'i iH2~ 0so that for consistency we require that the retarded and advanced responses be equal; this is of course m."ldatory if (1 5) is to be viewed as an effective static interaction. In practice the imaginary part of the response function always tends to zero at zero frequency anyway [15]. For the diagrams of fig. 1 a further simplification results from the fact that the two vertices are joined by two otherwise disconnected conduction electron propagators. Hence
~= kk'pp' oO' Ml~'
(p~IH~ Ik'o') (p'~'IH~ Iko) T ~ Gk,~.p,~(z+)Gk'o, p'.'(z-) 2
218
D.A. Smith/New mechanism .for magnetic anisotropy
where G,k," , ~ p u(z).. = ((Cka Icpu}) and the sum is over imaginary fermion frequencies. In general, T times such a sum of a function F(z) whose only singularities are poles and is of O(z-V), v > 1, at infinity, i~ just the sum of the resid aes o f f ( z ) F(=) at the poles of F, where f(z) is the Fermi.Dirac distribution.
_._. ;ffw dipolar-exd~ange term This contribution is represented by diagram (a) and its mirror image, giving
In second order perturbation theery there is no possible problem with enerb~y denominators. The requfred trace ! over spin variables is of the form Tr(A-sB. s) =-~ A" B, so that an anisotropic interaction of type ( I ) i s produced. In the free-electron approximation ( k I v v l l7k ' l = f d r e x p { - i ( k - k ' ) . r }
VV
l
7'
(16)
where the plane waves are normalised to unit volume (and hence N is atoms per unit volume). Hence the anisotropy tensor can be written as
"
x=tZuB)2N l
(2t+l)S t VRv R fdr'
gt(r')
Ir'- R['
(17)
where
fk'-fk P,(k'k')exp{-i(k-k').r}. kk' e k , - ek
g/(r)=.__~l ~ N2
(18)
The integration variable has been changed so that r' - R = r where R = R12,• r' is the electron coordinate with respect to ~he second spin carrying the exchange vertex. Now the asymptotic form of(17) i:~'entirely determined from the behaviour of the integr~md of (17) near r' = R as the integral can be written as
jd
RI - 47r
-~
r2gl(r)dr + f rgl(r)dr
(~9)
R ~ u s the dipole interaction is really required only for large r; this occurs because it can bc written as a double gradient of a 1/r potential. It should not be treated as a short-ranged interaction. "Hie asymptotic form of (18)is found to be
gt(r)--- 6nZp(_l) t c°s(2kFr) (2kF r) 3
C'.O)
from which the asymptotic form of (19) can be found on integrating by parts. Hence the anisotropy tensor is ~.r,en by' ~_ ~.,~u ~a) with ¢, = U t t l l t l B=(S/~B)2(2kF) 3
~
/=0,1,2
(--1)l+l(2l.l).llO .
(~ 1 )
We have used a free electron bav, d e k = k2/m *, Z is metallic valency and P = 3/4el,,.
2. 3. The term of second order it, spin-orbit coupling Di~,gram (b) of fig. 1 also leacts to an animtropic interaction. A general expression for the anisotmpy ~ensor is
D.A. Smith/New mechanism [or magneth" anisotropy I¢2 K-
( 2 1 + 1 ) ( 2 i' + l ) J i J I,
4kl: 4N 4
219
~ E ( k p k ' p ' ) k X p k' X p' kk'pp'
x Pi(k.p')e t, (p.k')exp{ i[(p'-k).R l + (,~-p).R,~ +(p-k').R 2 +(k'-p')"R~I},
(22)
where
E(kpk'p')= ek-eP (ek_ek,_i6)(ek_ep._i6)- (e_ek,_iS)(ep_ep,_iS) e k , - e ~,
(ek,-ek÷i~)(ek,--ee+i~)
(ep,-ek+i6)(ep,-ep-~i6)
(23)
'
and 6 can be a positive or negative infinitesimal. The identity
T r ( A ' s B ' s C ' s D . $ ) - - ~ ( A " B C . D + A . D B . C .... A . C B , D ) ,
(24)
where the vectors A to D are assumed to commute, ha: been used to remove electron spin variables: R a. and R~ are the sites of the two spin-orbit coupling vertices. We shall only consider the case when the magnetic i,~ns are themselves the sole source of spin-orbit scattering. Taken literally, this implies that only light metallic hosts with negligible spin-orbit potentials at ion cores can be considered. But since the anisotropy expression ( l ) is uniaxial, it may be applied to magnetic ions in heavier host metals of cubic symmetry because the resulting cubic anisotropy from host ion cores will be a much smaller effect. in this case the spin.orbit interaction ( 1 3 ) a t magnetic ion sites should be regarded as a scattering potential in the presence of the host. The k sums in (,.,.) may be replaced by integrals in the usual way and may be separated into radial and angular parts. Thus the angular integrals are specified by the tensor
(RIR.,R3R4)..
(It')
_
×k×pk'×p'exp{
I
.....i ( k . R I .... P . R 2 + k ' . R
~
p'.R4);,.
(25)
whose evaluation is given In an appendix. There a~e tw,~ casts to t.'(,llsid~2I, siilcc the t w ~ Spill-¢~lhll verl iCeS may belong to the same magnetic ion or to different ions. (i) Re, = R O = R I or R 2 : Denote the anisotropy tensors by Kll and K22 respectively. Then the required angular integral is C(II')iR 0 O R ) which is non-zero only if I' = I. As this ,~bject is iu~variant on interchange of k' and p and its anisotropic part is also invariant under interchange ot'k and p' we find that K II
K-,:, = ""
w2m'3JI
4N a-
l
l
t
(p'R)
J
(~,n2)4 •" ,
[//(kR)i; X
G 12l+ ............1...)JI ...... [. kl: ddd f~? 0
..............
t
it(kR) lt(P R) /~/~
0
~3k . 3p3p . 3dkdk . . dpdp ,k-' k' -' ,6,,k-
p K
Y~(l) ( ~'-,
+ ( a b o v e k .......k', p' ~.... p)
where R = R 12' {ii) R = R l, R = R 2 or vice vets;,: Here we require C (it')(R 0 ..... R O) which is non.zero only when I = !' = !. It is invariant on interchanging p and p' but ch,~ilges sign on interchanging k and k'. It fi,llows that KI2 = K2I = 0 ,
(27)
D.,4. Smith/New mechanisw~for mt~.netic anisotropy
220
Q
Fig. 2. Site geometry for Dyzaloshinskii-Moriya exchange. as the rest of the multiple :integrand is even under the combined transformation. "l'he radial integrals in (26) contain an unexpected feature. Although all the infinite integrals may appear t ~ be dive~ent, those containing spherical Bessei functions are convergent in the Cesaro sense and for these the :behaviour of the integrand at large k is not essential for the asymptotic behaviour of the integral at large R. But here we have additional integrals not containing Bessel fimctions which, if taken literally, are obviously divergent. In fact, these int~ra!s will be "'cured" bv restoring the radial k dependences of the matrix elements (10) and (13), which will exhibit the necessary cut-off behaviour. We do not intend to carry out this modification, which: would praduce considerable algebraic inconvenience and is not necessary for order o f magnitude estimates. Thus the integral
o
k2
~ gl (k) + ig2(k ), - p-~ - i6
(28)
musl be interpreted as indicated, where g2(k) ~ 5! rrk2 and gl (k) depends upon the precise form of cut-~,ff bul will :~atisfy lgl (k)i ~< k 2 where k c estimates the cut-off radius of the matrix elements. We expect that k c t> k F All other integrals are standard. The final integral from 0 to k F yields an asymptotic expansion for large R on integrat ing by parts. Our final result for the anisotropy is of the form ( 1 - 3 ) . The constants are B = 7r3r/W2Jl(J 0 - 2 J 1 + 5J2)Zp3~an q~= 2glg2/(g 2 --gl ),
(29)
',vher,~. ~ = 1 + (gi/g2) 2 andg I a n d g 2 are evaluated at k = k F.
Z. D:.va~.¢,sh inskii-?~h~ro,a exchange Diagram ~e} of fig. 1 and a correspt :~ding diagram with vertices 1 and 2 interchange :l both contribute equally ~o give the fonn
= D. (S 1 × S 2),
(30)
where D - - 2N2k 2.
II"
,
2i,i)(2t'+l)JzJ),
× exp(-i[(k-k'l.r-x.
1
kk'p
k × k'
(k+k').R +p'R]}
{
(ek_ek,~ek_ 6,r) (ek._¢Ic)(ek,__ep)
(Cl_ek)(ep_Ek,) j
(31)
!
:~nd R = R12 as before, r --. Ro~ - 7 (R! + R2) where R~ is the spin-orbit site. The spin trace required in this in.stance is i
Tr(A. sB" sO" ~.) =-~ A " (B X C). No attemp~ will be made to evaluate (3,1) explicitly, although there is no real obstacle to doing so. We consider
D.A. Smith/Newmechanismfor magneticanisotropy
221
only the symmetry property o(R,
- r) - - o(R,r),
(32)
satisfied by (31). Thus if there are two spin-orbit sites specified by vectors r and - r , i.e. possessing inversion symmetry about the mid-point between the two spin sites, then D will be zero unless the two spin-orbit coupling constants are different. An important example occurs when the spins themselves carry spin-orbit couplings (r = iI R). In this case the two spins must be inequivalent to produce a Dzyaloshinskii-Moriya exchange o)upling; if the s-d exchange and spin-orbit couplings are both different then D involves quantities like
w(1)j~ l)Jl(,2) _ w(2)~ 2)Jl!l)' where superscripts denote sites.
3. Single-ion anisotropy
The method of calculating anisotropic pair interactions given in section 2.1 will also yield single-ion anisotropies if the two spins in question are considered to be identical. Since the derivation assumed that the spins could be treated as classical objects, it becomes open to the objection that when the two spins are the same the result may not be quantum-mechanically correct. However, any additional terms arising from the non-commutivity of the spin with itself would yield effective Hamiltonians linear in the spin S; such terms are not expected in the absence of an external magnetic field as the original Hamiltonian exhibits time-reversal symmetry. One further consideration is that since the energy of interaction is now a self-energy a factor of one-half must be supplied in eq. (15). Of all the diagrams in fig. 1 the simplest kind, represented by (a), contains vertices at the spin site only and so is not expected to give anisotropy. Diagrams which lead to Dyzaloshinskii-Moriya exchange cannot contribute either and so the dominant contribution is from (b). Thus the anisotropy tensor/: defined in (4) is actually given by eq. (22) with R 1 = R 2 = 0 and an extra factor of one-half. The special case considered in ref. [9] is when the two spin-orbit vertices are at the same site: this is appropriate for small concentrations of spin-orbit scatterers. By setting R~ = R~ = R the anisotropy for this situation can be written as _
w2 2kF4N4 11' X Re
(2t+l)(21'+l)Jlsr (27r2) 4
kl
. f fffk2k'2p2p'2dkdk'dpdp' o 0
~
(ek-ep) (ek-e k, -iS) (ek-e v, - i 8 ) '
(33)
after permuting k vectors to achieve a common fermi factor and using the symmetries of the angular integral given in the appendix. For the terms with l = 1' = 2, this expression is identical with that of Smith and Haberkern except for the imaginary infinitesimals a~ppearing in the denominator, which were inadvertently omitted. From the appendix it follows that F has the form (5), and the asymptotic form for large R is lr-~ ,e I f ' i X
r tt~
A cos (4kFR) ......... ,
(2kFR)7
where A = - 36~r3(25 J~ + 2 0 J 1 J 2 + 3j2)w2p3Z, which is exactly twice the result that is implicit in ref. [9].~ A formal corrigenda for tef. [91 will appear in the appropriate journal.
222
D.A. Smith/New mechanism .for magnetic anisotropy
If the two spin-orbit centres are at different sites (Ra 4: R#), the general expression (22) shows that the angular integnd [equir~ is C *'~ll'~(Ra R~ RoR¢~). Explicit expressions for anisotropy will not be given, but the methods of the appe.,z.dix are sufficient to show that the angular form of the anisotropy must be as given in eq. (6).
4. Pfiscussion The magnetic materials to which our calculations are relevant must be of lower than cubic symmetry, since we have considered only those effects which arise in low-order perturbation theory, which yield uniaxial anisotrol?ies. They must. of course, also be local-moment bearing materials. A possible class of materials is that of dilute alloys of 3d elements: these are rather hard to come by since the requirement of non-cubic symmetry dictates a polyvaler~t host metal and 3d moments in such hosts are empirically associated with high Kondo temperatures or spin fluctuation temperatures. The only example known to the au~Ihor is ZnMn which shows anisotropy in the magnetic susceW, ibility [I 6]. ]'he hcp structure should provide a sensitive test for our proposed mechanism for single-ion anisotropy because the sum of contributions given by (6) over all twelve Zn neighbours of a Mn ion (assumed substitutional) is a multiple of the unit tensor for an ideal hcp lattice, which is formed by close packing of spheres. If the observed anisotropy arises in th~s way then it should depend sensitively on the departure of the c / a ratio from its ideal value 1.633. It is interesting to note that the host metal in which anisolropy has been seen is one of the very few hcp structures with c/a markedly different from 1.633. Another way of achieving a non-cubic distribution of spin-orbit centres aroun,!~a magnetic ion is to alloy with a small concentration of heavy impurities such as Pt [9]. In more concentrated loc2 loment alloys, i.e. spin glasses and mictomagnets, a random distribution of spinorbit centres about a magnetic ion may be achieved by virtue of the neighbouring magnetic ions, provided that they display a large ~pin-orlbit scattering potential. Hence single ion anisotropy can be expected in these materials too. ~n addition, ani~otropic exchange may also be expected whenever pairs of magnetic ions are in proximity. Now bulk magnetocrystalline anisotropy is not expected in these materials, at least in the absence of field-cooling, since there is no spontaneous magnetization: this appears to be so in practice [17]. But this is not to say that local anisotropy, on an at ~mic scale, is absent. Detailed Mossbauer evidence for local anisotropy in AuFe alloys has been presented by Window [18]. who gives specific statements about the easy directions of clusters of nearestr~e~ghbour Fe spins. For clu~i;ters of two and three Fe atoms, his conclusions may be summarised by saying that the directions betweeta p~Lirsare hard" thus in a cluster of two Fe atoms the (parallel) spins are perpendicular to the hne of centres and i~ a cluster of three the spins are normal to the plane of the cluster, i.e. (111). These statements are alll compatible x~ith pair anisotropy of the pseudodipolar type, viz. as given by (2), as stated by Window. They are al,~o compatible with single-ion anisotropy of type (6), even including the term G'/~ 1 X/~ 2 ~ 1 × 1~2 if' G' is positive. The e,~plicit calculaT:ion of anisotropy coefficients that appears in this paper may appear rathei crude since freeelectron wave functions have been used throughout and the radial k dependence e.f matrix elements has often been ignored. It is imporl ant to u~aderstand in what way this affects the results. The basic anisotropy law - the angui!ar dependence and the inverse 19ower law of the asymptotic form for large R - is just a consequence of the particular process of perturbalion thet~ry and is independent of the particular choice of Bloch functions. In very simple c~lculations of this type the nu ~aerical coefficient of the coupling is also wave-function independent in the sense tllat the matrix element.,, are reqLired only on the fermi surface. This is true for the RKKY exchange constant but not for any p~ocesses c~nsidered here, so the plane-wave values of coupling constants given in (21), (29) and (34) should be regarded as an order-of-magnitude estimate. To extend the calculation of anisotropic couplings to ne~rnei~_hbour spacings is a much more difficult exercise, reqt, iring the use of realistic Bloch wave-functions and the inclusio~ of potent!al scattering at the sites in question to all orders of perturbation theory. This last step is essential. since tl~e phase of the cosine term (3) is sensitive to higher-order scatterings and can often be expressed in terms of the appropriate phase shifts [19]. The rather depressing history of attempts to calculate hyperfine fielcLs in the Heusler alloys [20] mzly be attributed to these considerations,, and that is why no attempt has been made I~e~e :~) give even o~der-of-magnitude estimates for anisotropic couplings in near-neighbour situations.
D.A. Smith/New mechanism ]'ormagnetic anisotropy
223
Anisotropic exchange between S-state ions has also been considered from a somewhat different point of view by Levy [21 ], who has shown in a general way how it may arise from the spin-orbit splitting of the conduction band. Our explicit results are of course compatible with Levy's, but they also apply to disordered situations where the conduction band spin-orbit potential does not have the translational symmetry of the lattice, in fact, spin-orbit disorder can produce much larger anisotropies than otherwise expected by lowering the relevant symmetry to the point where uniaxial anisotropy appears. One last comment may avoid confusion over a minor point. In this paper pair anisotropy has been written in the form S 1 • R R. S 2 to indicate the anisotropic component of the total interaction arising from a given diagrarn. Isotropic components may also occur but have been omitted since they are dominated by the existing RKK Y interaction. Thus the pair anisotropy could equally well be written in the form S ! • R R . S 2 - -f S 1 • $2 which is correct for a coupling of dipolar type. A
^
5. Conclusions In metallic materials with localised S-state magnetic moments, there exist mechanisms for single-ion and pair anisotropy which do not involve ionic excited states, in contrast to the normal mechanisms in insulators. In particular, the pair anisotropy can be long-ranged and oscillatory after the fashion of the isotropic (RKKY)exchange coupling. Dzyaloshinskii-Moriya exchange between two spins is also possible in the presence of additional spinorbit centres of low symmetry. The predicted interactions appear to be consistent with the rather limited experimental evidence for anisotropy in these materials.
Acknowledgments I wish to thank Dr. Brian Window for drawing my attention to the pseudodipolar interaction as originally conceived by van Vleck [22] and for communicating his recent work. I am grateful to the referee for supplying the reference to the work of P.M. Levy.
Appendix: The integral (25) The angular integral of eq. (25) can be obtained in closed form by first making replacements such as
k-~ iVRI, for the component vectors of the two vector products. Each exponential function may then be expanded in terms of spherical harmonics and the angular integrals performed. Thus (25) can be written in the form
C kpk'p' (ll') (R IR2R3R4 ) = - F kp' (/) (R1R4)XXtr(I')(R2R3), -pk'
(AI)
where the meaning of the double product is most easily illustrated by its application to dyadics, i.e.,
(a B) XX (C D) = (A × C) (B X D). A correspoltding definition for general second-rank tensors is implied. He~e one has
F(l)kp(R 1R2)=V 1 V2/l( kR 1)ll(PR2)Pl( • 1 " ~ 2), where 11is tile spherical Bessel function of order I and PI the Legendre polynomial. More explicitly,
(A2)
22a
D A. Smith~New mechanism for m~,cnetic anisotropy k F'/t~ (R IR2) = kp j~(kRi)]~(pR2)Pt(R I • R2) ,~lR2 +R--2Ih(kRl) h(PR2)P[(Rt p
P "'
"/~2)(R2
"R~) ~1 (/}1 - ~"/~2'}2)
R2"/} *~I)R'
(A3)
h I~:Rl)h(p~2 ) .., + .........R-f-R-' ...... tQ l-- (PI'+ k I • R2P/") (kl k I + R2k2) + e/"R2k I + Rl. R2(P/'+ R1./~2p i ) k l k 2 }, where I is the unit second-rank tensor, and primes denote differentiation. Some part icular values of this tensor are required in the text. These are
'
t
'
F':;{R,~) =-~ h(kR)Ji(PR)P'( l) l + kp]](kR)]](pR)-~-~, h(kR)~,(pR)P[(1)
F~:p(OO)=
.~.
t=l-
0,
) RR,
(Aa)
t~l,
(A5)
/l(kR)\ . .
•) R R ,
1=1;
O, 1 ¢ 1 ,
(A6)
and F(OR) may be obtained from the identity (I) till R.~) 07 Fkpx-_ = c(l) -pk~,..2Rl ).
(A7)
By combinin:z these functions with the aid of identities such as TXXt = 2 1 . ~ ' X X A B = A B X X I
=A.BI
-EA,
(A8)
~hich are d.--ived fronl the definition indicated in (A2), we find that
,...,H, "' t~ kpk'p IRRRR) = R 14 ~RR fjl(kR)pRji:(pR)k' l R:I'(k'R)/t(P'R)t'I'(1 ) + kR((kR)]r(pR)/r(k'n)p'R]/(p'R)p[,(l)_
2]l(kR)/r(pR)/r(k'R)Jl(p'R)pl'(l)p[,(l)]
'
J
and
C(II') kv&.l,.(R ,""
0- R O)•
+ ~sotroplc tensor).
(A9)
kk'pp' Rt~ ( Jl(kR) . ]l(k"R) \ '9 k-----R ] 2 ( k ; R ) - ] 2 ( k R ) - - ~ R ~ ) . t
1 =, = 1;
O,
l or l' :/: 1.
(A10)
The~e functio3~s display the symmetry properties
C kpk' (~') 9" (RRRR) = C k'p'kp ~;'t) (RRRR) = ~pkp'k' r,(t'l) (RRRR),
(All)
~~1~ ( R O - - R O ) = Ckp,~.,~(RO-RO)= (11) (|1) , ( R O - R O ) . C kpk'v. - C k'zkp
(A~2)
and
D.A. Smith/New mechanism .for magnetic anisotropy
225
References 111 A.D. Caplin and J.B. Dunlop, J. Phys. l" (metal Physics) 3 (1973) 1621. 121 H. Brooks, Phys. Rev. 58 (1940) 909. [31 G.C. Fletcher, Proc. Phys. Soc. A, LXVII (1954) 505. I4,1 K. Yosida, J. Appl. Phys. 39 (-1968)511. I51 H. Watanabe, Prog. Th. Phys. (Kyoto) 18 (1957)405. I61 M.A. Ruderman and C. Kittel, Phys. Rev. 96 (1954) 99. [71 K. Yosida, Phys. Rev. 106 (1957) 893. I81 N. Bloembergen and T.J. Rowland, Phys. Rev. 97 (1955) 1679. [91 D.A. Smith and G.P. Haberkern, J. Phys. F (Metal Physics) 3 (1973) 856. IlOl S. Schultz, M.R. Shanabarger and P.M. Platzman, Phys. Rev. Letters 19 (1967) 749. !111 T. Moriya, in "Magnetism," eds. G.T. Rado and H. Suhl (Academic Press, New York 1963) vol !, p. 85. 1121 J.R. Schrieffer and P.A, Wolff, Phys. Rev. 149 (1966) 491 and J.R. Schrieffer, J. Appl. Phys. 38 (1967) 1143.
[131 A.O.E. Animalu, Phil. Mag. 13 (1966) 53. 114] A.A. Abrikosov, L.P. Gorkov and I. Ye. Dzyaloshinskii, "Quantum Field Theoretical Methods in Statistical Physics", 2nd [151 1161 !171 I181
ll9l [2Ol 1211 1221
edition (Pergamon Press, Oxford 1965). P.C. Martin, in "lVlany-Body Physics", Les Houches, eds. C. de Witt and R. Balian (Gordon and Breach. New York 1968) p 37. P.L. Li, F.T. Hedgcock, W.B. Muir and J.O. StriJm-Olsen, Phys. Rev. Letters 31 (1973) 29. T. lwata, K. Kai, T. Nakamichi and M. Yamamoto, J. Phys. Soc. Japan 28 (1970) 582. B. Window, Phys. Rev. B6 (1972) 2013 and in "Amorphous Magnetism" eds. H.O. Hooper and A.M. de Graaf (Plenum Pre~s. New York 1973) p. 229. A. Blandin and J. Friedel, J. Phys. Radium (Paris) 20 (1959) 160. D.J.W. Geldart, C.C.M. Campbell, P.J. Pothier and W. Leiper, Canadian J. Phys. 50 1'972) 206. P.M. Levy, SoKd State Comm. 7 (1969) 1813. J.H. van Vleck, Phys. Rev. 52 (1937) I 178.
NEW MECHANISMS FOR MAGNETIC ANISOTROPY IN LOCALISED S-STATE MOMENT MATERIA,I.~ D.A. SM~H Physics Department, Monash University, Clayton, Victoria 3168, Australia
Anisotropie versions of the familiar (RKKY) indirect exchange betweei) localised S-state magnetic moments in metals can arise in a number of ways. There are two basic mechanisms, viz. (i) from anisotropic s-d interactions such as dipolar coupling and (ii) in higher orders of perturbation theory with the conduction electron spin-orbit interaction. Applications to particular materials are briefly considered. 1. Introduction This paper is primarily concerned with local magnetic anisotropy, such as anisotropic exchange or single ion amsotropy, in metallic local moment materials. These are metallic elements, alloys or compounds for which the origin of magnetism resides in localised magnetic moments, which will be restricted herein to "spin only" me. ments, i.e, with g near 2. Examples are Gd, the Heusler alloys, Au2Mn, Znl3Mn [ 1] and indeed all dilute inter'. metallic compounds and alloys of the 3d transition elements provided that good moments exist and]or the tern. perature range of interest is well above the Kondo temperature. The microscopic origin of bulk magnetocrystalline anisotropy in the ferromagnetic transition metals is an old problem of band theory which has been discussed by Brooks [2] and Fletcher [3]. Here anisotropy arises frcm the corabined effects of 3d spin-orbit interaction and spin-split energy bands. This kind of mechanism is not likely to wo:k for local-moment metals becat:se the spin-orbit coupling cannot act directly on an orbitally non-degenerate moment and the conduction band will be only weakly spin-split. On the other hand, the conventional mechanism for single 3d ion anisotropy in insulators [4] requires orbitally degenerate excited crystal field states of the ion which are mixed into the crystal field ground state by spin-orbit coupling. A similar mechanism works even for S-state ions if higher terms (possibly of different configurations) which are themselves crystal field split are mixed intc, the S-state by magnetic interactions [5]. But for 3d moments in metals our present understanding, at least in the opinion of this author, is not yet at the stage where the lack of orbital degeneracy (as evidenced by the g factor) can confidently be ascribed to quenching by crystalline fields, so that any discussion of anisotropy arising from ex. cited crystal field states would be purely speculative. For the purposes of this paper we shall leave this question open by regarding all local moments as S-state ions, thus implying that no orbital moment is required at any stage of the calculations. The s-d exchange model, which treats the 3d ion as a localised spin, is sufficient for most of the processes conside,red here, which are best specified in diagrammatic form. Fig. ! lists the Feynmann diagrams for anisotropic indirect exchange between two localised spins S 1 and S 2 via the conduction electrons. In looking for an anis,gtropic interaction of the simplest firm = -- S'1 • X" $2,
(I)
(where K must be other than a +multiple of the unit tensor) a necessary condition is that there exists a virtual + process giving the interaction S I S 2 whicll changes the quantum numbers of both spins by + 1. In all diagrams an electron-hole pair is emitted from one spin and absorbed by the other. The dominant process, which gives rise to the usual RKKY exchange [6,7], has an s.d exchange interaction of the form S.,~ (s being a conduction electron
D.A. Smith~New mechanism for magnetic anisotropy
M¢1
M1
T
MZ
M¢I
~t't
M1
(a) "~,/"X""P"~
M2
M 2,.1 (b)
I
M~I
~ ,.
MI
V
(c)
Us
215
~
M2
M2¢'1
(d)
m2 (e)
(f)
Fig. I. Feynmann diagrams considered for anisotropic spin cuuplings. Full lines indicate conduction electron propagators and the dotted lines are included to indicate the initial and final states of the two Iocalised spins. • indicates an s-d exchange vertex (10), n the dipolar interaction (1 ! ), O the orbital interaction (i 2) and X indicates spin-orbit coupling (I 3) at some other site. Some diagrams have been spin-labelled to indicate anisotropic exchange, etc. as discussed in the text.
spin) at each vertex: this particular vertex conserves total spin and so gives indirect interactions S~ S~, S~S~ etc. consistent with the isotropic form S t • S 2. The simplest way to obtain anisotropic exchange is to introduce a vertex which does not conserve spin such as the dipolar spin-spin interaction; fig. l a is spin-labelled for an S~s + vertex. This mechanism has been given for the analagous case of nuclear spins by Bloembergen and Rowland [8]. Alternatively, if spin-orbit centres are admitted which scatter conduction electrons then Bloch states of opposite spin are admixed and an S+s" vertex is effectively converted into S+s z (fig. l b). This process also yields single-ion anisotropy if the two spins are replaced by just one, and has already been given by Smith and Haberkern [9]. Both processes yield a uniaxial anisotropy tensor of the form K = K(R) R/~, (2) in which the axis in question is the direction of the vector R joining the two spins or, in the case of single-ion anisotropy, the vector from the spin to the spin-orbit centre. Thus pair anisotropy is not directly re:ated to the crystal axes but to the directions defined by pairs of magnetic ions. The distinction would of course be spurious if this anisotropy were limited to nearest neighbours, but this is not the case. For both mechanisms the asymptotic form for large R is K ( R ) ~-
B cos(2kFR + 9)
,
(3)
(2kFR)3 where k v is the fermi radius, which is long-ranged and oscillatory with tile same power law and wavelength (though not necessarily the same phase) as the RKKY interaction itself. Some consequences of this result arc ex_l___a ltt _, ,Eric: L . ~tlu -_~ o f , Ltttt; - papt:i. ptult:u
The magnitude of B can vary quite widely. For the process of fig. l a it is most easily s~ecifie:l by saying that the coefficient of R - 3 i.e. B/(2kI:) 3, is smaller than the corresponding coefficient (2/~B) '~ of the direct dip,;le. dipole interaction by a factor of the order of J2P, where ,It is the lth partial wave component of s-d exchange and p is the electronic density of states at the fermi level per spin per atom. This factor, which is typically of the order of 0.07 for 3d moments, is exactly the expected ratio of the indirect interaction to the direct one. For fig. l b the value of B is of the order of JI J 2 w2p3 where w is the strength of the spin.orbit coupling, assumed to be centred at the magnetic ions. Hence this process will dominate the direct dipole interactioa for large w; ?. more preclise estimate for Mn in Cu (for which J 2 = - 0.5 eV, p -- 0.15 eV -I and JI possibly = - 0.05 eV) is that w > 0.5 eV. In
21 b
D..4. Smith/New mechanism for magnetic anisotropy
fact the appropriate value of w in QtMn is 0. i 2 eV from spin scattering off Mn sites, from e.s.r, measuremertts by Schultz, Shanabarger and Ptatzman [ 10], so that rather heavy spin-orbit scattering is required. Of the remaining diagrams in fig. !, (c) is an example of a process which cannot give an anisotropic interaction + + at all as it cannot be spin labelled to yield an S 1 S 2 term. (d) involves the orbital magnetic interaction (12) and spin-.~rbit interaction, which may be symbolised by S. I and I. s respectively; this diagram will give anisotropic exdlange as in (!) but its magnitude witl be smaller than the contribution from (a) by a factor of wp. (e) involves the combination 5! • s I.s 52 "s which, on tracing out the conduction spin uroduct, gives Dzyaloshinskii and Me,riya exchange [ 11 ] of the form D. (51 X 52). For D to be non-zero the distribution of spin-orbit centres around the two spin sites must be of low symmetry. In particular, when the spin-orbit centres are at the spin sites themselves D will be zero unless the spin-orbit coupling constants (or s-d exchange constants) at the two sites are different. Graph (~ is similar but will give a smaller interaction because of the dipolar vertex. l'he .same processes can in principle also give rise to single-ion anisotropy, which can formally be extracted by treating the two spins as just one. Then anisotropy can be created only by spimorbit centres at different sites. Thins has already been considered elsewhere [9] but the results will be summarised for completeness. Graph (e) does not contribute, but the dominant contribution is from (b), giving a short-rang,: uniaxial anisotropy of the form
~'
=
-
S.
(4)
F-S.
if there is only one spin-orbit centre close to the magnetic ion then F= F(R ) [~/t,
(5)
where R is the vector between the two sites. For two centres at positions R 1 and R, from the magnetic ion additional tensor forms, not considered in ref. [9], arise and the total contribution looks like F = "r:'l Rlkl + F 2 / ~ 2 R 2 + G(~'lk2 +/~2/~1) + G ' / ~ 1 X k2 1~1X/~2 ,
(6)
where the coefficients are functions ofR ! and R 2. The general case of n neighbouring centres would be described bv a superposition of pairs. There is :ao long-range effect here analagous to the R K K Y interaction; thus F(R) has an asymptotic form proportional to R-7 so that the net anisotropy tensor will be dominated by nearest neighbour contributions.
2. The calculation of anisotropic interactions 2. 1. ~ e ' Hamiltonian and met,hod oj' calculation Consider two S-state ions of the 2d series located at positions R 1 and R 2 in a free-electron-like metal. The ions will be represented by localised spin:; 1~1 and S 2 which interact with the conduction electrons by various "s.d" int, factions, viz. exchange, dipolar s?iLn-spin and magnetic spin-orbital interactions. The conduction electrons also suf :er spin-orbit scattering at sites laaelled by Ra, which will be specified further in due course. Thus the cornpie, e Hamlltonian is it = H 0 + H',+ W,
(7)
wh,:re
Hc, =
+
ko
q.
Cko,
(8)
is the con Juction electron tern: in second quantised form, /f' - *'1:'~ + Hdir. + Hc,rb"
(9)
D.A. Smith/New mechanism.formagneticanisotropy
217
is the sum of the s-d interactions, and W is the spin-orbit interaction. It is sufficient to specify H ' and W by their one-electron forms. Thus for a spin S at the origin of coordinates we write
(klHexlk),
=
~ (2/+ - N 1 t=0,1,2
1)Jl(k,k')Pl(k.k' ) S.s.•
Hdip(r ) "- - (2/.tB) 2 S- (V V 1 ) ' s Horb(r ) ,,. (2#B)2
S'I
r3
(!0)
(r large),
( ! I)
(r large),
(12)
where #B is the Bohr magneton and s and I are the spin and orbital angular momenta of the electron at r. ( 11 ) and (12) are the classical forms valid when the radius r exceeds that of the 3d shell; the exact expressions, which depend on the internal atomic structure of the ion, will not be required. Of the three exchange coefficients, "/2 will dominate as it contains the covalent admixing term given by Schrieffer and Wolff [12]. The one-electron matrix element for a spin-orbit centre at the origin is taken as (klWlk')
-
iw(k,k')
k X k'
N
k2F
"s,
(13)
which is a simplified version of Animalu's model potential [l 3]. The precise dependence of these matrix elements on k and k' is not crucial since it is the angular dependence which determines the form of indirect interactions; thus J/(k, k' ) and w(k, k' ~1will be replaced with their fermi surface values. Of the various ways of deriving indirect interactions, perhaps the simplest is in terms of linear response theory. which will give the energy tetra linear in S 1 and S 2 [7]. The advantage of this method is that it leads directly to an expression in terms of propagators, thus providing an automatic prescription for dealing with vanishing energy denominators. To formalise the method, write H I=-S
1.hI,
H 2=-S
2 . h 2,
(~'+~
where h I and h 2 are the respective conduction electron operators. H~ may be thought of as the static limit of a time-dependent perturbation; this perturbation induces a mean internal field (h2) = ((h2 JH'l ))0 + where ((A [B))z is the analytic continuation of the Fourier coefficient of the temperature Green function of A and B [14]. Here 0 + indicates zero frequency with a positive infinitesimal imaginary part, indicating the retarded response. By considering the torque that this field exerts on the second spin, the energy of interaction is found to be (]5)
~ = _ $ 2 . ( h 2 ) = ((H~ IH'1 )>0+" Moreover, a completely analogous argument treating 11'2 as a perturbation will lead to the same result with H ! interchanged. A hermicity relation for Green functions can now be used to write
H 2 and
((H 2 IU'1))0+ = ((H'i iH2~ 0so that for consistency we require that the retarded and advanced responses be equal; this is of course m."ldatory if (1 5) is to be viewed as an effective static interaction. In practice the imaginary part of the response function always tends to zero at zero frequency anyway [15]. For the diagrams of fig. 1 a further simplification results from the fact that the two vertices are joined by two otherwise disconnected conduction electron propagators. Hence
~= kk'pp' oO' Ml~'
(p~IH~ Ik'o') (p'~'IH~ Iko) T ~ Gk,~.p,~(z+)Gk'o, p'.'(z-) 2
218
D.A. Smith/New mechanism .for magnetic anisotropy
where G,k," , ~ p u(z).. = ((Cka Icpu}) and the sum is over imaginary fermion frequencies. In general, T times such a sum of a function F(z) whose only singularities are poles and is of O(z-V), v > 1, at infinity, i~ just the sum of the resid aes o f f ( z ) F(=) at the poles of F, where f(z) is the Fermi.Dirac distribution.
_._. ;ffw dipolar-exd~ange term This contribution is represented by diagram (a) and its mirror image, giving
In second order perturbation theery there is no possible problem with enerb~y denominators. The requfred trace ! over spin variables is of the form Tr(A-sB. s) =-~ A" B, so that an anisotropic interaction of type ( I ) i s produced. In the free-electron approximation ( k I v v l l7k ' l = f d r e x p { - i ( k - k ' ) . r }
VV
l
7'
(16)
where the plane waves are normalised to unit volume (and hence N is atoms per unit volume). Hence the anisotropy tensor can be written as
"
x=tZuB)2N l
(2t+l)S t VRv R fdr'
gt(r')
Ir'- R['
(17)
where
fk'-fk P,(k'k')exp{-i(k-k').r}. kk' e k , - ek
g/(r)=.__~l ~ N2
(18)
The integration variable has been changed so that r' - R = r where R = R12,• r' is the electron coordinate with respect to ~he second spin carrying the exchange vertex. Now the asymptotic form of(17) i:~'entirely determined from the behaviour of the integr~md of (17) near r' = R as the integral can be written as
jd
RI - 47r
-~
r2gl(r)dr + f rgl(r)dr
(~9)
R ~ u s the dipole interaction is really required only for large r; this occurs because it can bc written as a double gradient of a 1/r potential. It should not be treated as a short-ranged interaction. "Hie asymptotic form of (18)is found to be
gt(r)--- 6nZp(_l) t c°s(2kFr) (2kF r) 3
C'.O)
from which the asymptotic form of (19) can be found on integrating by parts. Hence the anisotropy tensor is ~.r,en by' ~_ ~.,~u ~a) with ¢, = U t t l l t l B=(S/~B)2(2kF) 3
~
/=0,1,2
(--1)l+l(2l.l).llO .
(~ 1 )
We have used a free electron bav, d e k = k2/m *, Z is metallic valency and P = 3/4el,,.
2. 3. The term of second order it, spin-orbit coupling Di~,gram (b) of fig. 1 also leacts to an animtropic interaction. A general expression for the anisotmpy ~ensor is
D.A. Smith/New mechanism [or magneth" anisotropy I¢2 K-
( 2 1 + 1 ) ( 2 i' + l ) J i J I,
4kl: 4N 4
219
~ E ( k p k ' p ' ) k X p k' X p' kk'pp'
x Pi(k.p')e t, (p.k')exp{ i[(p'-k).R l + (,~-p).R,~ +(p-k').R 2 +(k'-p')"R~I},
(22)
where
E(kpk'p')= ek-eP (ek_ek,_i6)(ek_ep._i6)- (e_ek,_iS)(ep_ep,_iS) e k , - e ~,
(ek,-ek÷i~)(ek,--ee+i~)
(ep,-ek+i6)(ep,-ep-~i6)
(23)
'
and 6 can be a positive or negative infinitesimal. The identity
T r ( A ' s B ' s C ' s D . $ ) - - ~ ( A " B C . D + A . D B . C .... A . C B , D ) ,
(24)
where the vectors A to D are assumed to commute, ha: been used to remove electron spin variables: R a. and R~ are the sites of the two spin-orbit coupling vertices. We shall only consider the case when the magnetic i,~ns are themselves the sole source of spin-orbit scattering. Taken literally, this implies that only light metallic hosts with negligible spin-orbit potentials at ion cores can be considered. But since the anisotropy expression ( l ) is uniaxial, it may be applied to magnetic ions in heavier host metals of cubic symmetry because the resulting cubic anisotropy from host ion cores will be a much smaller effect. in this case the spin.orbit interaction ( 1 3 ) a t magnetic ion sites should be regarded as a scattering potential in the presence of the host. The k sums in (,.,.) may be replaced by integrals in the usual way and may be separated into radial and angular parts. Thus the angular integrals are specified by the tensor
(RIR.,R3R4)..
(It')
_
×k×pk'×p'exp{
I
.....i ( k . R I .... P . R 2 + k ' . R
~
p'.R4);,.
(25)
whose evaluation is given In an appendix. There a~e tw,~ casts to t.'(,llsid~2I, siilcc the t w ~ Spill-¢~lhll verl iCeS may belong to the same magnetic ion or to different ions. (i) Re, = R O = R I or R 2 : Denote the anisotropy tensors by Kll and K22 respectively. Then the required angular integral is C(II')iR 0 O R ) which is non-zero only if I' = I. As this ,~bject is iu~variant on interchange of k' and p and its anisotropic part is also invariant under interchange ot'k and p' we find that K II
K-,:, = ""
w2m'3JI
4N a-
l
l
t
(p'R)
J
(~,n2)4 •" ,
[//(kR)i; X
G 12l+ ............1...)JI ...... [. kl: ddd f~? 0
..............
t
it(kR) lt(P R) /~/~
0
~3k . 3p3p . 3dkdk . . dpdp ,k-' k' -' ,6,,k-
p K
Y~(l) ( ~'-,
+ ( a b o v e k .......k', p' ~.... p)
where R = R 12' {ii) R = R l, R = R 2 or vice vets;,: Here we require C (it')(R 0 ..... R O) which is non.zero only when I = !' = !. It is invariant on interchanging p and p' but ch,~ilges sign on interchanging k and k'. It fi,llows that KI2 = K2I = 0 ,
(27)
D.,4. Smith/New mechanisw~for mt~.netic anisotropy
220
Q
Fig. 2. Site geometry for Dyzaloshinskii-Moriya exchange. as the rest of the multiple :integrand is even under the combined transformation. "l'he radial integrals in (26) contain an unexpected feature. Although all the infinite integrals may appear t ~ be dive~ent, those containing spherical Bessei functions are convergent in the Cesaro sense and for these the :behaviour of the integrand at large k is not essential for the asymptotic behaviour of the integral at large R. But here we have additional integrals not containing Bessel fimctions which, if taken literally, are obviously divergent. In fact, these int~ra!s will be "'cured" bv restoring the radial k dependences of the matrix elements (10) and (13), which will exhibit the necessary cut-off behaviour. We do not intend to carry out this modification, which: would praduce considerable algebraic inconvenience and is not necessary for order o f magnitude estimates. Thus the integral
o
k2
~ gl (k) + ig2(k ), - p-~ - i6
(28)
musl be interpreted as indicated, where g2(k) ~ 5! rrk2 and gl (k) depends upon the precise form of cut-~,ff bul will :~atisfy lgl (k)i ~< k 2 where k c estimates the cut-off radius of the matrix elements. We expect that k c t> k F All other integrals are standard. The final integral from 0 to k F yields an asymptotic expansion for large R on integrat ing by parts. Our final result for the anisotropy is of the form ( 1 - 3 ) . The constants are B = 7r3r/W2Jl(J 0 - 2 J 1 + 5J2)Zp3~an q~= 2glg2/(g 2 --gl ),
(29)
',vher,~. ~ = 1 + (gi/g2) 2 andg I a n d g 2 are evaluated at k = k F.
Z. D:.va~.¢,sh inskii-?~h~ro,a exchange Diagram ~e} of fig. 1 and a correspt :~ding diagram with vertices 1 and 2 interchange :l both contribute equally ~o give the fonn
= D. (S 1 × S 2),
(30)
where D - - 2N2k 2.
II"
,
2i,i)(2t'+l)JzJ),
× exp(-i[(k-k'l.r-x.
1
kk'p
k × k'
(k+k').R +p'R]}
{
(ek_ek,~ek_ 6,r) (ek._¢Ic)(ek,__ep)
(Cl_ek)(ep_Ek,) j
(31)
!
:~nd R = R12 as before, r --. Ro~ - 7 (R! + R2) where R~ is the spin-orbit site. The spin trace required in this in.stance is i
Tr(A. sB" sO" ~.) =-~ A " (B X C). No attemp~ will be made to evaluate (3,1) explicitly, although there is no real obstacle to doing so. We consider
D.A. Smith/Newmechanismfor magneticanisotropy
221
only the symmetry property o(R,
- r) - - o(R,r),
(32)
satisfied by (31). Thus if there are two spin-orbit sites specified by vectors r and - r , i.e. possessing inversion symmetry about the mid-point between the two spin sites, then D will be zero unless the two spin-orbit coupling constants are different. An important example occurs when the spins themselves carry spin-orbit couplings (r = iI R). In this case the two spins must be inequivalent to produce a Dzyaloshinskii-Moriya exchange o)upling; if the s-d exchange and spin-orbit couplings are both different then D involves quantities like
w(1)j~ l)Jl(,2) _ w(2)~ 2)Jl!l)' where superscripts denote sites.
3. Single-ion anisotropy
The method of calculating anisotropic pair interactions given in section 2.1 will also yield single-ion anisotropies if the two spins in question are considered to be identical. Since the derivation assumed that the spins could be treated as classical objects, it becomes open to the objection that when the two spins are the same the result may not be quantum-mechanically correct. However, any additional terms arising from the non-commutivity of the spin with itself would yield effective Hamiltonians linear in the spin S; such terms are not expected in the absence of an external magnetic field as the original Hamiltonian exhibits time-reversal symmetry. One further consideration is that since the energy of interaction is now a self-energy a factor of one-half must be supplied in eq. (15). Of all the diagrams in fig. 1 the simplest kind, represented by (a), contains vertices at the spin site only and so is not expected to give anisotropy. Diagrams which lead to Dyzaloshinskii-Moriya exchange cannot contribute either and so the dominant contribution is from (b). Thus the anisotropy tensor/: defined in (4) is actually given by eq. (22) with R 1 = R 2 = 0 and an extra factor of one-half. The special case considered in ref. [9] is when the two spin-orbit vertices are at the same site: this is appropriate for small concentrations of spin-orbit scatterers. By setting R~ = R~ = R the anisotropy for this situation can be written as _
w2 2kF4N4 11' X Re
(2t+l)(21'+l)Jlsr (27r2) 4
kl
. f fffk2k'2p2p'2dkdk'dpdp' o 0
~
(ek-ep) (ek-e k, -iS) (ek-e v, - i 8 ) '
(33)
after permuting k vectors to achieve a common fermi factor and using the symmetries of the angular integral given in the appendix. For the terms with l = 1' = 2, this expression is identical with that of Smith and Haberkern except for the imaginary infinitesimals a~ppearing in the denominator, which were inadvertently omitted. From the appendix it follows that F has the form (5), and the asymptotic form for large R is lr-~ ,e I f ' i X
r tt~
A cos (4kFR) ......... ,
(2kFR)7
where A = - 36~r3(25 J~ + 2 0 J 1 J 2 + 3j2)w2p3Z, which is exactly twice the result that is implicit in ref. [9].~ A formal corrigenda for tef. [91 will appear in the appropriate journal.
222
D.A. Smith/New mechanism .for magnetic anisotropy
If the two spin-orbit centres are at different sites (Ra 4: R#), the general expression (22) shows that the angular integnd [equir~ is C *'~ll'~(Ra R~ RoR¢~). Explicit expressions for anisotropy will not be given, but the methods of the appe.,z.dix are sufficient to show that the angular form of the anisotropy must be as given in eq. (6).
4. Pfiscussion The magnetic materials to which our calculations are relevant must be of lower than cubic symmetry, since we have considered only those effects which arise in low-order perturbation theory, which yield uniaxial anisotrol?ies. They must. of course, also be local-moment bearing materials. A possible class of materials is that of dilute alloys of 3d elements: these are rather hard to come by since the requirement of non-cubic symmetry dictates a polyvaler~t host metal and 3d moments in such hosts are empirically associated with high Kondo temperatures or spin fluctuation temperatures. The only example known to the au~Ihor is ZnMn which shows anisotropy in the magnetic susceW, ibility [I 6]. ]'he hcp structure should provide a sensitive test for our proposed mechanism for single-ion anisotropy because the sum of contributions given by (6) over all twelve Zn neighbours of a Mn ion (assumed substitutional) is a multiple of the unit tensor for an ideal hcp lattice, which is formed by close packing of spheres. If the observed anisotropy arises in th~s way then it should depend sensitively on the departure of the c / a ratio from its ideal value 1.633. It is interesting to note that the host metal in which anisolropy has been seen is one of the very few hcp structures with c/a markedly different from 1.633. Another way of achieving a non-cubic distribution of spin-orbit centres aroun,!~a magnetic ion is to alloy with a small concentration of heavy impurities such as Pt [9]. In more concentrated loc2 loment alloys, i.e. spin glasses and mictomagnets, a random distribution of spinorbit centres about a magnetic ion may be achieved by virtue of the neighbouring magnetic ions, provided that they display a large ~pin-orlbit scattering potential. Hence single ion anisotropy can be expected in these materials too. ~n addition, ani~otropic exchange may also be expected whenever pairs of magnetic ions are in proximity. Now bulk magnetocrystalline anisotropy is not expected in these materials, at least in the absence of field-cooling, since there is no spontaneous magnetization: this appears to be so in practice [17]. But this is not to say that local anisotropy, on an at ~mic scale, is absent. Detailed Mossbauer evidence for local anisotropy in AuFe alloys has been presented by Window [18]. who gives specific statements about the easy directions of clusters of nearestr~e~ghbour Fe spins. For clu~i;ters of two and three Fe atoms, his conclusions may be summarised by saying that the directions betweeta p~Lirsare hard" thus in a cluster of two Fe atoms the (parallel) spins are perpendicular to the hne of centres and i~ a cluster of three the spins are normal to the plane of the cluster, i.e. (111). These statements are alll compatible x~ith pair anisotropy of the pseudodipolar type, viz. as given by (2), as stated by Window. They are al,~o compatible with single-ion anisotropy of type (6), even including the term G'/~ 1 X/~ 2 ~ 1 × 1~2 if' G' is positive. The e,~plicit calculaT:ion of anisotropy coefficients that appears in this paper may appear rathei crude since freeelectron wave functions have been used throughout and the radial k dependence e.f matrix elements has often been ignored. It is imporl ant to u~aderstand in what way this affects the results. The basic anisotropy law - the angui!ar dependence and the inverse 19ower law of the asymptotic form for large R - is just a consequence of the particular process of perturbalion thet~ry and is independent of the particular choice of Bloch functions. In very simple c~lculations of this type the nu ~aerical coefficient of the coupling is also wave-function independent in the sense tllat the matrix element.,, are reqLired only on the fermi surface. This is true for the RKKY exchange constant but not for any p~ocesses c~nsidered here, so the plane-wave values of coupling constants given in (21), (29) and (34) should be regarded as an order-of-magnitude estimate. To extend the calculation of anisotropic couplings to ne~rnei~_hbour spacings is a much more difficult exercise, reqt, iring the use of realistic Bloch wave-functions and the inclusio~ of potent!al scattering at the sites in question to all orders of perturbation theory. This last step is essential. since tl~e phase of the cosine term (3) is sensitive to higher-order scatterings and can often be expressed in terms of the appropriate phase shifts [19]. The rather depressing history of attempts to calculate hyperfine fielcLs in the Heusler alloys [20] mzly be attributed to these considerations,, and that is why no attempt has been made I~e~e :~) give even o~der-of-magnitude estimates for anisotropic couplings in near-neighbour situations.
D.A. Smith/New mechanism ]'ormagnetic anisotropy
223
Anisotropic exchange between S-state ions has also been considered from a somewhat different point of view by Levy [21 ], who has shown in a general way how it may arise from the spin-orbit splitting of the conduction band. Our explicit results are of course compatible with Levy's, but they also apply to disordered situations where the conduction band spin-orbit potential does not have the translational symmetry of the lattice, in fact, spin-orbit disorder can produce much larger anisotropies than otherwise expected by lowering the relevant symmetry to the point where uniaxial anisotropy appears. One last comment may avoid confusion over a minor point. In this paper pair anisotropy has been written in the form S 1 • R R. S 2 to indicate the anisotropic component of the total interaction arising from a given diagrarn. Isotropic components may also occur but have been omitted since they are dominated by the existing RKK Y interaction. Thus the pair anisotropy could equally well be written in the form S ! • R R . S 2 - -f S 1 • $2 which is correct for a coupling of dipolar type. A
^
5. Conclusions In metallic materials with localised S-state magnetic moments, there exist mechanisms for single-ion and pair anisotropy which do not involve ionic excited states, in contrast to the normal mechanisms in insulators. In particular, the pair anisotropy can be long-ranged and oscillatory after the fashion of the isotropic (RKKY)exchange coupling. Dzyaloshinskii-Moriya exchange between two spins is also possible in the presence of additional spinorbit centres of low symmetry. The predicted interactions appear to be consistent with the rather limited experimental evidence for anisotropy in these materials.
Acknowledgments I wish to thank Dr. Brian Window for drawing my attention to the pseudodipolar interaction as originally conceived by van Vleck [22] and for communicating his recent work. I am grateful to the referee for supplying the reference to the work of P.M. Levy.
Appendix: The integral (25) The angular integral of eq. (25) can be obtained in closed form by first making replacements such as
k-~ iVRI, for the component vectors of the two vector products. Each exponential function may then be expanded in terms of spherical harmonics and the angular integrals performed. Thus (25) can be written in the form
C kpk'p' (ll') (R IR2R3R4 ) = - F kp' (/) (R1R4)XXtr(I')(R2R3), -pk'
(AI)
where the meaning of the double product is most easily illustrated by its application to dyadics, i.e.,
(a B) XX (C D) = (A × C) (B X D). A correspoltding definition for general second-rank tensors is implied. He~e one has
F(l)kp(R 1R2)=V 1 V2/l( kR 1)ll(PR2)Pl( • 1 " ~ 2), where 11is tile spherical Bessel function of order I and PI the Legendre polynomial. More explicitly,
(A2)
22a
D A. Smith~New mechanism for m~,cnetic anisotropy k F'/t~ (R IR2) = kp j~(kRi)]~(pR2)Pt(R I • R2) ,~lR2 +R--2Ih(kRl) h(PR2)P[(Rt p
P "'
"/~2)(R2
"R~) ~1 (/}1 - ~"/~2'}2)
R2"/} *~I)R'
(A3)
h I~:Rl)h(p~2 ) .., + .........R-f-R-' ...... tQ l-- (PI'+ k I • R2P/") (kl k I + R2k2) + e/"R2k I + Rl. R2(P/'+ R1./~2p i ) k l k 2 }, where I is the unit second-rank tensor, and primes denote differentiation. Some part icular values of this tensor are required in the text. These are
'
t
'
F':;{R,~) =-~ h(kR)Ji(PR)P'( l) l + kp]](kR)]](pR)-~-~, h(kR)~,(pR)P[(1)
F~:p(OO)=
.~.
t=l-
0,
) RR,
(Aa)
t~l,
(A5)
/l(kR)\ . .
•) R R ,
1=1;
O, 1 ¢ 1 ,
(A6)
and F(OR) may be obtained from the identity (I) till R.~) 07 Fkpx-_ = c(l) -pk~,..2Rl ).
(A7)
By combinin:z these functions with the aid of identities such as TXXt = 2 1 . ~ ' X X A B = A B X X I
=A.BI
-EA,
(A8)
~hich are d.--ived fronl the definition indicated in (A2), we find that
,...,H, "' t~ kpk'p IRRRR) = R 14 ~RR fjl(kR)pRji:(pR)k' l R:I'(k'R)/t(P'R)t'I'(1 ) + kR((kR)]r(pR)/r(k'n)p'R]/(p'R)p[,(l)_
2]l(kR)/r(pR)/r(k'R)Jl(p'R)pl'(l)p[,(l)]
'
J
and
C(II') kv&.l,.(R ,""
0- R O)•
+ ~sotroplc tensor).
(A9)
kk'pp' Rt~ ( Jl(kR) . ]l(k"R) \ '9 k-----R ] 2 ( k ; R ) - ] 2 ( k R ) - - ~ R ~ ) . t
1 =, = 1;
O,
l or l' :/: 1.
(A10)
The~e functio3~s display the symmetry properties
C kpk' (~') 9" (RRRR) = C k'p'kp ~;'t) (RRRR) = ~pkp'k' r,(t'l) (RRRR),
(All)
~~1~ ( R O - - R O ) = Ckp,~.,~(RO-RO)= (11) (|1) , ( R O - R O ) . C kpk'v. - C k'zkp
(A~2)
and
D.A. Smith/New mechanism .for magnetic anisotropy
225
References 111 A.D. Caplin and J.B. Dunlop, J. Phys. l" (metal Physics) 3 (1973) 1621. 121 H. Brooks, Phys. Rev. 58 (1940) 909. [31 G.C. Fletcher, Proc. Phys. Soc. A, LXVII (1954) 505. I4,1 K. Yosida, J. Appl. Phys. 39 (-1968)511. I51 H. Watanabe, Prog. Th. Phys. (Kyoto) 18 (1957)405. I61 M.A. Ruderman and C. Kittel, Phys. Rev. 96 (1954) 99. [71 K. Yosida, Phys. Rev. 106 (1957) 893. I81 N. Bloembergen and T.J. Rowland, Phys. Rev. 97 (1955) 1679. [91 D.A. Smith and G.P. Haberkern, J. Phys. F (Metal Physics) 3 (1973) 856. IlOl S. Schultz, M.R. Shanabarger and P.M. Platzman, Phys. Rev. Letters 19 (1967) 749. !111 T. Moriya, in "Magnetism," eds. G.T. Rado and H. Suhl (Academic Press, New York 1963) vol !, p. 85. 1121 J.R. Schrieffer and P.A, Wolff, Phys. Rev. 149 (1966) 491 and J.R. Schrieffer, J. Appl. Phys. 38 (1967) 1143.
[131 A.O.E. Animalu, Phil. Mag. 13 (1966) 53. 114] A.A. Abrikosov, L.P. Gorkov and I. Ye. Dzyaloshinskii, "Quantum Field Theoretical Methods in Statistical Physics", 2nd [151 1161 !171 I181
ll9l [2Ol 1211 1221
edition (Pergamon Press, Oxford 1965). P.C. Martin, in "lVlany-Body Physics", Les Houches, eds. C. de Witt and R. Balian (Gordon and Breach. New York 1968) p 37. P.L. Li, F.T. Hedgcock, W.B. Muir and J.O. StriJm-Olsen, Phys. Rev. Letters 31 (1973) 29. T. lwata, K. Kai, T. Nakamichi and M. Yamamoto, J. Phys. Soc. Japan 28 (1970) 582. B. Window, Phys. Rev. B6 (1972) 2013 and in "Amorphous Magnetism" eds. H.O. Hooper and A.M. de Graaf (Plenum Pre~s. New York 1973) p. 229. A. Blandin and J. Friedel, J. Phys. Radium (Paris) 20 (1959) 160. D.J.W. Geldart, C.C.M. Campbell, P.J. Pothier and W. Leiper, Canadian J. Phys. 50 1'972) 206. P.M. Levy, SoKd State Comm. 7 (1969) 1813. J.H. van Vleck, Phys. Rev. 52 (1937) I 178.