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Submitted on 1 Jan 1978
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LONGITUDINAL SPIN SUSCEPTIBILITY OF A
FERROMAGNETIC METAL
J. Callaway, Avhishek Chatterjee
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supp/&ment au no 8, Tome 39, aotit 1978, page C6-772
LONGITUDINAL SPIN SUSCEPTIBILITY OF A FERROMAGNETIC METAL
J. Callaway and A.K. Chatterjee
Department o f Physics, Louisiana S t a t e U n i v e r s i t y , Baton Rouge, Lousiana 70808, U.S.A.
R6sum6.- On a obtenu une expression pour la susceptibilitd magndtique longitudinale d'un mdtal fer- romagngtique. L'interaction Blectron-glectron est reprgsentge dans 1'Hamiltonien par un potentiel qui est one fonction simple de la densitd dlectronique.
Abstract.- An expression for the longditudinal magnetic susceptibility of a ferromagnetic metal is obtained using a Hamiltonian containing an exchange correlation potential which is an ordinary function of density.
+
Band calculations based on local exchange Finally U is a spin operator and S i s a unit vector
correlation potentials have been quite successfulin in the direction of magnetization. describing the ground state properties of ferroma-
gnetic 3d transition metals /I/. It is of conside- In order to calculate the susceptibility, we
rable interest to investigate the extent to which add to (1) the interaction between a spin and an linear response functions can be obtained using on external time and position dependent magnetic field the same basis. The principal advantage of this ap-
proach is consistency. The electron interaction is included in susceptibility calculations in the sane approximation used for the band calculation. It is not necessary to graft an extra electron interaction term onto the single particle Hamiltonian in order to obtain results which have the expected form, and the expressions do not involve undetermined para-
meters.
A general procedure for the calculation of the static, unfirorm susceptibility in the local
The calculation of the response can be performed by ordinary time dependent perturbation theory. Care must be taken to insure that the calculation is self-consistent ; that is, the dependence of the potentials V. and Vf on density must be properly included. In the case of the transverse spin sus- ceptibility, which was obtained previously 141, it is necessary to allow for a rotation of
S.
Here we consider the longditudinal spin susceptibility only. A detailed account of our procedures will be pu- blished elsewhere, and we present just the essen- tial results. Since the general expressions becomedensity, approximation has previously been given by very lengthy when local field effects are included Vosko and Perdew / 2 / . We make here the assumption we will neglect local fields in this brief report.
that the exchange-correlation interaction is re-
presented in the single particle Hamiltonian by an The magnetic susceptibility is given by the explicit function of the local spin density. One formula
example of such a potential is that due to von Barth
(r++r++
4aT+rt) and Hedin 1 3 1 . Our Hamiltonian has the form
X
a 1 + (T++r+) (a+c) + (l'+-l'+) (b+d) + 4r+
r+(ac-bd)+
+
+H =
-
v2
+ VO(r) + Vf (r) 6.Ein which
ru
(a = f , J.) is given by in which V contains the interaction of an electron0 + -+ +
with all the nuclei of the system the average elec- 1 NnU(k)-Nb(k+p) trostatic repulsion of the electrons, and the spin l ' n
C+
Elu(k+p)-EnU(k)+-in + -++
average of the exchange correlation potential, Vxc.The qaantity Vf is the difference potential which leads to magnetic order,
( 2 ) The matrix elements and energies refer to Bloch
functions,
II
and n are band indices. The otherquantities a, b, c, d are averages of derivatives of the potentials, aVxcu
1
U < - > b = - ua' 3 ~ ~ 'In (5), U, U' =
*
I, and < > indicates a volume average over the unit cell. When local fields are included,a, b, c, d above as well as theru
must be replaced by matrices on a basis of reciprocal lat- tice vectors and the expressions become modified because the matrices need not commute. Somewhat si-milar expressions for the dielectric function and for cross response functions (spin-charge and charge-spin) can also be obtained.
If one makes the (drastic ) additional ap-
1
proximation that a P
-
C = - V (a constant), and 2sets b d P 0, equation (3) reduces to the expres-
sion obtained from a Hubbard model Hamiltonian by Izuyama, Kim, and Kubo 161. Our results show an es- sential difference, however, from that work. In the limit of a strong ferromagnet
(r+
= 0, say) we ob- tainwhich does not simply reduce to
r+
unless we make the crude approximation mentioned above. Thus the longditudinal susceptibility of a strong ferroma- gnet is not just the non-interacting susceptibility.We can also consider the limit of a para- magnet, in which p+ = p+ = p/2, T+ =
r+.
Then the parameters b, and d vanish and we havein which X)'( = 2I' is the usual non-interacting susceptibility, and we have set
then we find
in which the integral is taken over a unit cell of volume R. Finally we can easily show from (7) and
(8) that in the limit of a free electron gas (and p = 0 , fd = 0) that
X
reduces to the Hartree-Fock resultwhich is exact in the high density limit. Thus in a sense, our approximations interpolate between the free electron gas and the Hubbard model.
This work was supported in part by the U.S. National Science Foundation.
References
/l/ Wang, C.S. and Callaway, J., Phys. Rev.
B15
(1977) 298/2/ Vosko, S.H. and Perdew, J.P., Can. J. Phys.
53
(1975) 1385131 Von Barth, U. and Hedin, L., J. Phys. (1972) 1629
141 Callaway, J. and Wang, C.S., J. P ~ Y s .
5
(1975) 2119151 Callaway, J. and Chatterjee, A.K., submitted to J. Phys. F.
161 Izuyama, T., Kim, D.J. and Kubo, R.J., Phys. Soc. Japan
18
( 1 963) 1025so as to reproduce the usual Stoner formula. If we use the Kohn-Sham (exchange only) potential forV
