Download Simple Theory of the Magnetic Properties of Rare

VOLUME
PHYSICAL REVIEW LETTERS
26, NUMBER 12
*Work supported by the U. S. Air Force Office of
Scientific Research, Grant No. 68-1586, and by the
Advanced Research Projects Administration through
the Cornell Materials Science Center.
C. B. Duke and C. W. Tucker, Jr. , Phys. Rev. Lett.
28, 1168 (1969); Surface Sci. 15, 281 (1969). For discussion of structure analyses within the spirit of the
s-wave model see C. W. Tucker, Jr. , and C. B. Duke,
Surface Sci. 28, 411 (1970), and 24, 81 (1971); C. B.
Duke and G. E. Laramore, Phys. Rev. B 2, 4765, 4788
(1970).
C. B. Duke, J. R. Anderson, and C, W. Tucker, Jr. ,
Surface Sci. 19, 117 (1970).
~R. O. Jones and J. A. Strozier, Jr. , Phys, Rev. Lett.
22, 1186 (1969); J. A. Strozier, Jr. , aud B. O. Jones,
Phys. Hev. Lett. 25, 516 (1970).
J. B. Pendry, J. Phys. C: Proc. Phys. Soc., London
2, 1215, 2278, 2288 (1969).
G. Capart, "Dynamical Calculations of LKED Intensities at Clear Copper (001) Surface (to be published).
R. M. Stern and F. Balibar, Phys. Rev. Lett. 25,
1338 (1970).
'V. Hoffstein and D. S. Boudreaux, Phys. Rev. Lett.
25, 512 (1970).
Simple Theory
22 MwRm 1971
K. Kambe, Surface Sci. 20, 21$ (1970).
P. J. Jennings and K. G. McRae, Surface Sci. 28,
868 (1970).
~OF. Joua, IBM J. Bes. Develop. 14, 444 (1970).
~~B. I. Lundqvist, Phys. Status Solidi 32, 273 (1969).
'2J. I. Gersteu, Phys. Bev. B 2, 8457 (1970).
Duke and Laramore, Ref. 1.
All energy scales quoted are measured with respect
to the vacuum level, unless otherwise specified.
' Our definition of A. (=h, ) is the same as that defined in Ref. 2 and is twice that defined in the first
paper of Ref. 1.
J. L. Beeby, J. Phys. C: Proc. Phys. Soc., London
1, 82 (1968).
~'B. W. Holland and R. W. Hannum, to be published.
J. B. Pendry, private communication.
J. W. D. Connolly, in Proceedings of the Symposium
on Atomic, Molecular, and Solid-State Theory and
Quantum Biology, edited by P. Lowdin (Interscience,
New York, 1969), p. 807.
S. Y. Tong and T. N. Rhodin, to be published.
2~M. G. Lagally and M. B. Webb, Phys. Rev. Lett. 21,
1888 (1968).
E. G. McHae, J. Chem. Phys. 45, 8258 (1966).
of the Magnetic Properties of Rare-Earth Metals
I.. M.
Department
Falicov and C. E. T. Goncalves da Si1va~
of Physics, t University of California, Berheley, California 94720
(Received 18 January 1971)
A simple
theory for the magnetism
of the rare-earth metals is presented.
It is based
on assuming strongly correlated, well-defined f-electron states of an atomiclike character which are weakly hybridized with conduction-band states. This hybridization plays
the role of an indirect exchange interaction. A model calculation is presented: It yie1ds
for various values of the two relevant parameters ferromagnetic, antiferromagnetic,
These arrangements are strongly dependent on the features
and helical arrangements.
of the conduction-electron band structure (Fermi surface). Order of magnitude esti-
mates give reasonable agreement for physical parameters
ed experiments.
We report here a simple theoretical model
which explains qualitatively the many possible
magnetic arrangements of the rare-earth metals.
In this model the usual roles of hybridization between f-like and conduction-band wave functions
and correlation effects between f-like electrons
are reversed, i.e., the correlation effects are
taken into account as the important, zeroth-order contribution to the energy and the f-stateconduction-band hybridization terms are included
as a perturbation.
The same zeroth-order Hamiltonian has been
used successfully' to explain the n-y phase transition in cerium metal. It assumes that the levels and the conduction band are not hybridized.
f
'
as determined from unrelat-
The conduction states are essentially noninteracting Bloch states derived from the 6s and 5d states
f
of the atom. The localized
states are atomiclike (or ioniclike to be more precise) in character and are strongly correlated to one another,
constituting on the mhole a well-defined manyelectron structure with mell-defined total angular
momentum J. The energetics of this structure is
such that, for all phenomena involved here, only
one electron can be either removed from an
shell and placed (really or virtually) in a conduction state or vice versa. If, for the sake of definiteness and simplicity, me assume that only
occupations with zero or one electron are permitted, or zero or one hole as well, corre-
f
f
f
715
Vor. &MAL 26, NUMszR 12
sponding to cerium and ytterbium, respectively,
then the f-electron system is represented in second quantization by
=g Q
X~
Za, ~a,
„,
where E is the one-electron energy eigenvalue,
i designates the ion site, and
is the magnetic
quantum number. Correlation, vvhich is the
strongest effect in this case, imposes the constraint that no two states with the same i can be
I
The conduction-band
occupied simultaneously.
electxons are represented by a texm
(2)
as usual, and no constraint exists in this case.
%Alen hybx'1dlzRtlon
18 tRkeQ into Recount
term in the Hamiltonian
is of the form
R Qew
must be considered.
It
This te1m, among other effects, co1x'elRtes 1Q
fourth order the magnetic moments of the elect1ons Rt vR1 ious sites and p1ovldes RQ 1nterRct1OQ
which gives rise to long-1Rnge magnetic ox'dex'-
f
iIlg.
In ox'der to show how the model allow&8 fox' a
large variety of magnetic arrangements, me have
considered some very simple examples. In one
such example %Pe chose J= pq
=+@~
berg Hamiltonian:
The results for 8 corresponding to the first three
sets of nearest neighbors, (&a, —,'a, -', a), (a, 0, 0),
and {a,a, 0), are shown in Table I for various values of eF =-e F/W and he =(eF-E)/W; g is expressed in units of
A positive vRlue 1nd1CRtes RntlferroIDagnetic
coupling and a negative g corresponds to a ferromag-
netic interaction. The general trend of the values
is such that (a) ferromagnetic interactions tend
to appeax for eF close to e1ther the top or the
bottom of the conduction band; and (b) antiferromagnetic coupling is prepondexant for eF close
to Q, i.e., in the middle of the band.
With the interaction (6) and the values of Table
I we have calculated chvss~caOy the spin axxangement which minimizes the total energy. The minimum energy arrangements are, except for an
arbitrary uniform rotation of all spina, of the
form
S(R; ) = S{sin0;, 0, cos9, ),
I
~, = —W cos(k„a/2) cos(k, a/2) cos(k, a/2)
(4)
g = (2b, e/a)(0,
0, 1).
TRMe I* MRgQetiC RllRQgemeQt
-Ik'&eF
urations of
&
„. =6
„N '"V, exp(-ik R, ),
~
is consistent
with the completely periodic
properties of the system and the neglect of spinorbit coupling in the model; in (5) V is the number of ions in the crystal and Vo has dimensions
which
for
VRrloUS COTlflg-
simple model.
W,
but otherwise also fx ee to vary. The hybridization matrix element was chosen to be
Vi,
3,
of energy.
The dispersion relation (4) corresponds to an
s-like tight-binding band in a body-centered-cubic stxucture, The fourth-oxder perturbation
correction of the hybridization Hamiltonian (3)
can be easily expressed in the form of a Heisen-
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PHYSICAI. RKVIK%' LKYTKRS
Vor. UMz 26, NUMsam 12
Values of 66) ale given Rlso 1n TRble I. A vRlue
of 60=0 corx'esponds to a complete ferromagnetic arrangement, 68=~ is R perfect Rntiferromagnet, Rnd an arbitrary 60 gives a helical a.rrangement with a pitch P =am/268. It can be seen
that a large variety of arrangements appears.
Several points are worth remarking:
(A) If in (2) e„ is taken to be a free-electron
dispersion relation proportional to k', 8 can be
proven to give a "Ruderman-Kittel" type of coupling
)~)=~p, V,')V '
man-Kittel" contact interaction" or the extended
itinerant electx on-exchange mechanism' responsible for the spin-density wave state of chromium. The exchange in this case is perfectly localized and restricted to f-electron-f-electron
interaction in the same ionic site; it is in this
sense a strongly repulsive "hard-core" interaction. The long-rRnge coupling between spins
is R one-electx"on effect, i.e., a small hybridization of the shell and the conduction states caused
by the lattice potential. The conduction-electron
dispersion relation, which defines the "polarizability" of the medium, plays in this case (as in
all the other analogous ones) the role of propagating to the spin-spin interactions the singularities due to the Fermi distribution.
(E) In our formulation the effect appears in
fourth-order perturbation theory because two
spins have to flip to intexact with one another,
f
is the density of states at the Fermi level.
Equations
(ll) and (12) are of the right or(B)
der of magnitude. If we use Coqblin and Blandin'8
a, nd
vRlue8q
ing
and pz
to cerium, de Gennes's pax'amI' (which is proportional to g) turns out to
corresponding
be
1"=5.8 eV A',
while de Gennes estimates it from experiment
to
be
(C) As expected from general considerations'
'
and fxom analogy to Cr metal, the details of the
Fermi surface are pax'amount in determining
xnagnetic arrangements.
In particular a perfectly "nesting" Fermi surface, such as in our example for eF =0, should strongly favor a pex'feet ant1ferromagnetlc ox'dex'1ng.
(D) It should be emphasized however that the
mechanism
responsible for magnetic properties
is different fx'om either the 'Ruder-
in oux' theory
ea, ch 8pln flip 1nvolves Rs usuRl two scRttex'-
processes ("in" and "out" or vice versa) between extended and localized states.
(6 (=-0.05 eV, eF-E, ~0.1 eV,
eter"
22 MwmcH 1971
"On a Conselho Nacional de Pesquisas (Brasil) Scholar ship.
(Work supported in pary by the National Science Foundation under Grant No. GP 13889.
K. N. H. Taylor, Contemp. Phys. 11, 423 (1970).
H. Bamirez aud L. M. Falicov, Phys. Rev. B (to be
published) .
3M. A. Huderman
and C. Kittel, Phys. Hev. 96, 99
(1954); C. Kittel, Quantum ZVseo~y of Solids (Wiley,
New York, 1963), aud in SOHd Stute Physics, edited
by H. Ehreureich, F. Seitz, aud D. Turnbull (Academic, New York, 1969), Vol. 22, p. l.
B. Coqblin and A. Blandin, Adv. Phys. 17, 281 (1968).
5P. G. de Gennes, Z. Phys. Radium 23, 510 (1962).
O. K. Anderson and T. L. Loucks, Phys. Hev. 167,
55~ (&968).
J. C. Kimball, Phys. Hev. 188, 588 (1969), and many
references cited therein.
8C. Herring, in Magnetism, edited by G. T. Hado and
H. Suhl (Academic, New York, 1966), Vol. 4.
ic, New York, 1966), Vol. 4.