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Submitted on 1 Jan 1979
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Anisotropic effects from spin-split bands
W. Fairbairn, A. Sharland, P. Strange
To cite this version:
W. Fairbairn, A. Sharland, P. Strange. Anisotropic effects from spin-split bands. Journal de Physique Colloques, 1979, 40 (C5), pp.C5-81-C5-82. �10.1051/jphyscol:1979530�. �jpa-00218947�
JOURNAL DE PHYSIQUE Colloque C5, supplkment au no 5, Tome 40, Mai 1979, page C5-81
Anisotropic effects from spin-split bands
W. M. Fairbairn, A. J. Sharland and P. Strange Dept of Physics, University of Lancaster, England
R6sumC. - Dans les terres rares mCtalliques l'interaction entre les moments IocalisCs se produit par I'interme- diaire des tlectrons de conduction et depend de la structure de bande. Dans un etat magnetiquement ordonne, la polarisation spatiale donne naissance 21 des structures de bande distinctes pour des Clectrons dans des Ctats de spin diffkrents. L'effet de cette polarisation a kt6 estimt pour l'Ctat ferromagnetique de Gd. Un modkle simple est utilisk pour Ctudier ces bandes skparkes par l'action des spins. On montre que l'interaction isotrope devient anisotrope. L'effet, bien que faible, est cependant susceptible d'altkrer les prkdictions concernant la tempkrature de Curie. La transition reste du second ordre.
Abstract. - In metallic rare earths the RKKY interaction between localized moments is mediated through the conduction electrons and depends on the band structure. In a magnetically ordered state the spatial polarization will produce different band structure for electrons in different spin states. The effect of this polarization has been estimated for the ferromagnetically ordered state of Gd. A simple model for the spin-split bands is used and shows that the isotropic interaction becomes anisotropic. The effect is not large, but it could alter predictions for the Curie temperature. The transition remains of second order.
That part of the Hamiltonian for a system contain- ing lattice ions of spin I and conduction electrons which relates to the interaction between a localized ion n (situated at R,) and the non-localized electrons is of the form [I]
where q = k' - k is the change in linear momentum of an electron scattered and
is the matrix element for the ion-electron scattering.
A second-order perturbation theory calculation results in the well-known RKKY expression for the energy of the indirect exchange interaction between two lattice ions n and m (at separation r) due to this coupling via the conduction electrons : it is [l]
where F(x) = (x cos x - sin x)/x4.
Expression (2) is obtained from (1) by assuming that the creation and annihilation operators, C + and C, are associated with electrons which can be considered as free, with effective mass m*, energy E = A2 k2/2 m*, and occupying spin-degenerate states in a conduction band which is full up to the Fermi
energy f i 2 k:/2 m*. A proper band-structure calcula- tion would give more accurate wave functions and these would permit a better estimate of the electron- ion scattering matrix element, which is treated in the RKKY derivation as a point interaction with energy-independent strength J. This possibility of obtaining a more realistic expression for the indirect exchange integral has been discussed further in [2].
Such a calculation in the case of magnetically ordered systems would allow also the occurrence of different energies for electrons in opposite spin states.
An immediate result of the electron operators in (1) having different effects for electrons of different spin is that the isotropic Heisenberg interaction (2) is changed to an anisotropic one. We estimate the size of this effect using the following model : the energy of a conduction electron is taken to be dependent on its spin so that there are spin-split bands which will, however, be chosen to be of free-electron cha- racter. The magnitude of the energy-splitting of the two bands will depend on the magnetic interaction between the electrons and the field due to the ordered ions. For a ferromagnetically ordered state this will be an internal field Bin, caused by the similarly aligned localized magnetic moments of all the rare earth ions. The outer electronic structure of gadolinium ions is (4Q7 with L = 0, S = 712 giving J = 712.
The orbital motion of these core electrons gives no contribution to the magnetic field, which is related to the magnetization of the localized moments through the alignment of the ionic spins S . If the z-axis is fixed (conveniently and realistically as the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979530
C5-82 W. M. FAIRBAIRN, A. J. SHARLAND AND P. STRANGE
c-axis of the crystal) then the energies of the spin- { Amn(Zmx Znx
+
Zmy Zny)+
Cmn Zmz Znz ] x split bands are given by E , = ti2 k2/2 m* ) yM,where M is the magnetization of the specimen, and X
9
( & ) 3 , (3) y is a constant.On evaluating the second-order contribution arising with A,, and C,, being functions of distance r and from (1) with the bands split as above, the ion-ion magnetization M. When M = 0, A,, = C,,. The interaction is found to be specific expressions for these coefficients are
b -
A,,,, =
(1
x sin x cosJm
dx+
x sin x exp(-1 J-
dx+
r4 0
+ J
x sin x cosJ F T P
dxj ,
(4a)a b +
em,
=f (J,,
x sin x cos x dx+
(4b)with b o - - k(0) r , A Z a 2 = 4 m * p b M r 2 , b t = b ? + a 2 , b : + b ? = 2 b i .
The experimental results for the spin-polariza- tion [3] indicate that the completely ordered state has a magnetic moment 0.63 p, per atom arising from the conduction electrons. This would require a change in appropriate Fermi momenta of the order of 7
%.
The corresponding values of the parameters in (4) are b+ = 1.07 b,, b- = 0.94 b,, a = 0.51 b, which indicate a considerable deviation from isotropy in the completely ordered state. The value for a corresponds to an energy splitting of the bands of approximately 0.8 eV, which is siifiilar to that indi- cated by optical experiments [2]. The value found by Mattocks and Young [4] is 0.82 eV. For temperatures nearer to the ordering temperature (smaller pola- rizations) the values of b+ and b - would be closer to b, and a would be smaller.The spins of the lattice ions now have the c-axis of the hcp crystal structure as preferred direction
for their cooperative orientation. This anisotropy is additional to any anisotropic effects due to a non- spherical Fermi surface, as discussed by Temple et al. [5]. However, the distance dependence of both effects is similar : the r-dependence of the RKKY interation (2) is given by F(2 k, r). Both anisotropies have r- variation, but different periods of oscillation.
The dependence of the free energy on magnetiza- tion M changes. A detailed investigation of the differences between the coefficients A,, and C,,, as given by (4a) and (4b), and their values in the iso- tropic case (2) shows that the energy of the system changes by terms which are quadratic in M. There are no linear terms in M, which is the ordering para- meter in these systems. The general form of the free energy will be unaltered ; the transition will remain of second order but the ordering temperature may change.
References
[l] K ~ T E L , C., Quantum Theory of Solidr (John Wiley & Sons, New York) 1963.
[2] HARMON, B. N., FREEMAN, A. J., Phys. Rev. B 10 (1974) 1979.
[3] ROELAND, L. W., COCK, G. J., MULLW, F. A., MOLEMANN, A. C., JORDAN, R. G., MCEWAN, K. A., J. Phys. F 5 (1975) L 233.
[4] MATTOCKS, P. G., YOUNG, R. C., J. Phys. F 7 (1977) 1219.
[5] TEMPLE, J. A. G., FAIRBAIRN, W. M., PICKETT, G. R., J. Phys.
F 7 (1977) 1039.
