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On the ferromagnetism of EuB6
Volume 34A. number 5
PHYSICS
length of the cell does not determine the scattering length in this case, since the S2-light leaves scattering volume oblique to the Si-direction alter a few millimeters (beam aperture inside the cell 0.2mm). Since at a given laser intensity the Si intensity is maximum at the end of the cell, it is clear that the process works preferably in this region. Both processes can be observed separately as noted probably for geometrical reasons. The increase of the S 2 wave in both cases is assumed as exp(glI~), where gis an gain factor, obviously different for the collinear and the oblique process, 1 is the corresponding scattering length and the pump intensity. One finds for the shorther cell at higher Ip the product 1. the
LETTERS
22 March 1971
same as before for the central emission, but increased by a factor v’! for the oblique emission. Once the threshold for the parametric process is exceeded, the depletion of the Si light would be so rapid that the collinear scattering can no longer occur (if I~is only a little larger than threshold).
References [1] U. Deserno and G.Nath, Phys. Letters 30A (1968) [21483. F. Aussenegg and U. Deserno. Optics Communications 2 (1970) 295. [31N. Bloembergen, Am.J. Phys. 35 (1967) p.989.
ON THE FERROMAGNETISM
OF EuB6
Z. FISK Department of Physics and The James Franck Institute The University of Chicago, Chicago, III. 60637, USA
Received 17 February 1971 An explanation for the ferromagnetism of semiconducting EuB6 as contrasted with the antiferromagnetism of metallic GdB6 is suggested, based on the Bloembergeb—Rowland exchange interaction.
2 (Jsf2/EF)S(S It ishexaborides a curious fact thatshow all the metallic rare earth which magnetic ordering are antiferromagnetic, while isostructural EuB 6, the only semiconducting hexaboride with a magnetic moment, is ferromagnetic [ii. Matthias has calledexplanation attention tofor this We suggest here a simple the[2]. difference. We make a comparison of semiconducting EuB 6 and7 metallic GdB6,[3]. bothExperiments Eu and Gd having configuration on the here of rate an depression f of the superconducting transition temperature of YB 6 by rare earth impurities suggest that the RKKY interaction is responsible for the interactions between localized f-electrons in the metallic rare earth hexaborides [4]. As9~int = -2J suming an interaction of the form 51 S. s between a localized spin S and a conduction electron molecular field approximation spin s, the RKKY and using interaction a freein electron conduction band predicts a paramagnetic Curie-Weiss temperature of
kBO
=
+
1)
-3irZ
~F(2kFr.ii ).
(1)
j~
E F is the Fermi energy, k F the magnitude of the
wave vector at the moments Fermi level and the= separation between local i and j.ofnj F(x) 4(xcosx - sinx). Z is the number electrons xin the conduction band: theory [5] and experiment [6] show that Z = 1 for the trivalent rare earth hexaborides. Bloembergen and Rowland [7] worked out a modification of the RKKY interaction for indirect exchange between local moments in semiconductors. For a parabolic valence band containing Z electrons with width band mucheffective less thatmass the band E and a conduction m*. gap tifèir result predicts
-
x ~
2(~i 2/2Et)S(s+ 1) 5i
-3rz
F(2k~r~ .) exp (_\‘2m*E r. ./Ji) gij
.
(2) 261
Volume 34A, numbers
PHYSICS
Here Et and kt are evaluated at the top of the valence band. Using the experimental 0 = -55°Kfor GdB6 2 from (i) taking only nearest [3], we evaluate J neighbors in the sum 5f over the simple cubic lattice of Gd in the hexaboride. This tends to underestimate J~ 1. Assuming J5f is the same for both Gd and Eu in their hexaborides, we predict using (2) that 0 = iO°K, in agreement with the experimentalt value equal 0to= the 9°K[3]. electron In mass (2) we and have Eg put =Z0.38 = 2. eV, m our experimental value for the thermal gap determined from high temperature electrical resistivity measurements. While ours is a very approximate treatment, it appears that the Bloembergen-Rowland interaction can account for both the magnitude and sign of Ofor EuB We note that De Graaf and Xavier [8-9] have6applied this mechanism in a study of the Eu chalcogenides. They point out that this mechanism might be expected to dominate the usual exchange mechanisms in insulators and semiconductors when direct exchange and the degree of covalency in the compound is low. EuB 6 meets both these conditions. It is interesting that the RKKY interaction using a free electron conduction band predicts antiferromagnetism for all cubic Bravais lattices of magnetic atoms for one conduction electron per magnetic atom [10~, while the Bloembergen-Rowland exchnage mechanism predicts
262
LETTERS
22March 1971
ferromagnetism in semiconductors with these lattices. We see, then, that in situations in which we have a cubic, semiconducting Eu cornpound (with Eu having an f7 configuration) and an isostructural Gd compound, we might often expect the former to be ferromagnetic (or have 0
>
0°K)and the latter to be antiferromagnetic.
We thank B. T. Matthias for many useful discussions. References [1] B. T. Matthias, T. H. Gebafle, K. Andres, E. Corenzwit, G. W. Hull and J. P. Maita, Science 159 (1968) 530. [2) B.T.Matthias, Phys. Letters 27A (1968) 511. [3] Yu. B. Paderno, S. Pokrzywnlcki and B. Stalinaki, Phys. Stat. Sol. 24 (1967) K73. (4] Z. Fisk, B. T. Matthias and E. Corenzwit, Proc. Nat. Acad. Sci. USA 64(1969) 1151. [5] H. C. Longuet-Higgens and M. deV. Roberts, Proc. (London) A224 (1954) J. 336. [6] Roy. R. W.Soc. Johnson and A. H. Daane, Chem. Phys. 38 (1963) 425. [7]N. Bloembergen and T. J. Rowland, Phys. Rev. 97 (1955) 1679. [8] A. M. DeGraaf and R. M. Xavier, Phys. Letters 18 225. [9] (1965) R. M. Xavier, Phys. Letters 25A (1967) 244. [10] D.C. Mattis, The theory of magnetism (Harper and Row, New York, 1967) chap. 7.
PHYSICS
length of the cell does not determine the scattering length in this case, since the S2-light leaves scattering volume oblique to the Si-direction alter a few millimeters (beam aperture inside the cell 0.2mm). Since at a given laser intensity the Si intensity is maximum at the end of the cell, it is clear that the process works preferably in this region. Both processes can be observed separately as noted probably for geometrical reasons. The increase of the S 2 wave in both cases is assumed as exp(glI~), where gis an gain factor, obviously different for the collinear and the oblique process, 1 is the corresponding scattering length and the pump intensity. One finds for the shorther cell at higher Ip the product 1. the
LETTERS
22 March 1971
same as before for the central emission, but increased by a factor v’! for the oblique emission. Once the threshold for the parametric process is exceeded, the depletion of the Si light would be so rapid that the collinear scattering can no longer occur (if I~is only a little larger than threshold).
References [1] U. Deserno and G.Nath, Phys. Letters 30A (1968) [21483. F. Aussenegg and U. Deserno. Optics Communications 2 (1970) 295. [31N. Bloembergen, Am.J. Phys. 35 (1967) p.989.
ON THE FERROMAGNETISM
OF EuB6
Z. FISK Department of Physics and The James Franck Institute The University of Chicago, Chicago, III. 60637, USA
Received 17 February 1971 An explanation for the ferromagnetism of semiconducting EuB6 as contrasted with the antiferromagnetism of metallic GdB6 is suggested, based on the Bloembergeb—Rowland exchange interaction.
2 (Jsf2/EF)S(S It ishexaborides a curious fact thatshow all the metallic rare earth which magnetic ordering are antiferromagnetic, while isostructural EuB 6, the only semiconducting hexaboride with a magnetic moment, is ferromagnetic [ii. Matthias has calledexplanation attention tofor this We suggest here a simple the[2]. difference. We make a comparison of semiconducting EuB 6 and7 metallic GdB6,[3]. bothExperiments Eu and Gd having configuration on the here of rate an depression f of the superconducting transition temperature of YB 6 by rare earth impurities suggest that the RKKY interaction is responsible for the interactions between localized f-electrons in the metallic rare earth hexaborides [4]. As9~int = -2J suming an interaction of the form 51 S. s between a localized spin S and a conduction electron molecular field approximation spin s, the RKKY and using interaction a freein electron conduction band predicts a paramagnetic Curie-Weiss temperature of
kBO
=
+
1)
-3irZ
~F(2kFr.ii ).
(1)
j~
E F is the Fermi energy, k F the magnitude of the
wave vector at the moments Fermi level and the= separation between local i and j.ofnj F(x) 4(xcosx - sinx). Z is the number electrons xin the conduction band: theory [5] and experiment [6] show that Z = 1 for the trivalent rare earth hexaborides. Bloembergen and Rowland [7] worked out a modification of the RKKY interaction for indirect exchange between local moments in semiconductors. For a parabolic valence band containing Z electrons with width band mucheffective less thatmass the band E and a conduction m*. gap tifèir result predicts
-
x ~
2(~i 2/2Et)S(s+ 1) 5i
-3rz
F(2k~r~ .) exp (_\‘2m*E r. ./Ji) gij
.
(2) 261
Volume 34A, numbers
PHYSICS
Here Et and kt are evaluated at the top of the valence band. Using the experimental 0 = -55°Kfor GdB6 2 from (i) taking only nearest [3], we evaluate J neighbors in the sum 5f over the simple cubic lattice of Gd in the hexaboride. This tends to underestimate J~ 1. Assuming J5f is the same for both Gd and Eu in their hexaborides, we predict using (2) that 0 = iO°K, in agreement with the experimentalt value equal 0to= the 9°K[3]. electron In mass (2) we and have Eg put =Z0.38 = 2. eV, m our experimental value for the thermal gap determined from high temperature electrical resistivity measurements. While ours is a very approximate treatment, it appears that the Bloembergen-Rowland interaction can account for both the magnitude and sign of Ofor EuB We note that De Graaf and Xavier [8-9] have6applied this mechanism in a study of the Eu chalcogenides. They point out that this mechanism might be expected to dominate the usual exchange mechanisms in insulators and semiconductors when direct exchange and the degree of covalency in the compound is low. EuB 6 meets both these conditions. It is interesting that the RKKY interaction using a free electron conduction band predicts antiferromagnetism for all cubic Bravais lattices of magnetic atoms for one conduction electron per magnetic atom [10~, while the Bloembergen-Rowland exchnage mechanism predicts
262
LETTERS
22March 1971
ferromagnetism in semiconductors with these lattices. We see, then, that in situations in which we have a cubic, semiconducting Eu cornpound (with Eu having an f7 configuration) and an isostructural Gd compound, we might often expect the former to be ferromagnetic (or have 0
>
0°K)and the latter to be antiferromagnetic.
We thank B. T. Matthias for many useful discussions. References [1] B. T. Matthias, T. H. Gebafle, K. Andres, E. Corenzwit, G. W. Hull and J. P. Maita, Science 159 (1968) 530. [2) B.T.Matthias, Phys. Letters 27A (1968) 511. [3] Yu. B. Paderno, S. Pokrzywnlcki and B. Stalinaki, Phys. Stat. Sol. 24 (1967) K73. (4] Z. Fisk, B. T. Matthias and E. Corenzwit, Proc. Nat. Acad. Sci. USA 64(1969) 1151. [5] H. C. Longuet-Higgens and M. deV. Roberts, Proc. (London) A224 (1954) J. 336. [6] Roy. R. W.Soc. Johnson and A. H. Daane, Chem. Phys. 38 (1963) 425. [7]N. Bloembergen and T. J. Rowland, Phys. Rev. 97 (1955) 1679. [8] A. M. DeGraaf and R. M. Xavier, Phys. Letters 18 225. [9] (1965) R. M. Xavier, Phys. Letters 25A (1967) 244. [10] D.C. Mattis, The theory of magnetism (Harper and Row, New York, 1967) chap. 7.