High-temperature weak ferromagnetism in a low-density free-electron gas.

The magnetic properties of the ground state of a low-density free-electron gas in three dimensions have been the subject of theoretical speculation and controversy for seven decades. Not only is this a difficult theoretical problem to solve, it is also a problem which has not hitherto been directly addressed experimentally. Here we report measurements on electron-doped calcium hexaboride (CaB6) which, we argue, show that-at a density of 7× 1019 electrons cm-3-the ground state is ferromagnetically polarized with a saturation moment of 0.07 µB per electron. Surprisingly, the magnetic ordering temperature of this itinerant ferromagnet is 600 K, of the order of the Fermi temperature of the electron gas.

14. Yagil, Y. et al. Experimental evidence for strong electron-phonon coupling to selected phonon modes in point contact spectroscopy. Physica C 250, 59±66 (1995  High-temperature weak ferromagnetism in a low-density free-electron gas The magnetic properties of the ground state of a low-density freeelectron gas in three dimensions have been the subject of theoretical speculation and controversy for seven decades 1 . Not only is this a dif®cult theoretical problem to solve, it is also a problem which has not hitherto been directly addressed experimentally. Here we report measurements on electron-doped calcium hexaboride (CaB 6 ) which, we argue, show thatÐat a density of 7 3 10 19 electrons cm 2 3 Ð the ground state is ferromagnetically polarized with a saturation moment of 0.07 m B per electron. Surprisingly, the magnetic ordering temperature of this itinerant ferromagnet is 600 K, of the order of the Fermi temperature of the electron gas.
The cubic hexaborides of the rare-earth elements have long attracted interest because of their wide variety of physical properties in spite of their simple crystallographic structures. These physical properties include very low work functions leading to the use of LaB 6 as a thermionic emitter 2,3 , dense Kondo behaviour and electric quadrupole ordering in CeB 6 (ref. 4), Kondo insulating properties in SmB 6 (ref. 5), and low-carrier-density ferromagnetism in the local moment system EuB 6 (refs 6, 7).
The host material for the present experiments is the divalent alkaline-earth hexaboride CaB 6 . The crystal structure of this material can be thought of as a simple cubic CsCl-type arrangement of B 6octahedra and metal ions. The early electronic structure cluster calculations of Longuet-Higgins and Roberts 8 found that the linked B 6 -network required 20 electrons for a`closed shell' electronic con®guration, indicating that the alkaline-earth hexaborides would be semiconductors. A study of the low-temperature properties of single crystals of SrB 6 , however, found a non-zero conductivity due to approximately 0.001 electrons per SrB 6 , and indications of the importance of electron±hole Coulomb effects 9 . More recent band structure calculations 10 show that hexaborides with divalent cations should be semimetals, with a small direct overlap of a primarily boron-derived valence band with a primarily alkaline-earth-derived conduction band at the X-point in the Brillouin zone.
Our experiments, including measurements of the electrical resistivity, the magnetic susceptibility and the magnetization, were performed on single crystals of CaB 6 doped with trivalent La; these crystals were grown from stoichiometric amounts of the hexaboride components in molten Al. On similarly prepared single crystals of Sr 1-x Ce x B 6 , we have veri®ed that Ce is incorporated stoichiometrically into the crystals by measuring the magnetic susceptibility and ®tting the data to a Curie±Weiss law with an effective moment m eff 2:54 m B per trivalent Ce ion. It is reasonable to assume that the neighbour of Ce in the rare-earth sequence, La, is also incorporated stochiometrically in SrB 6 and CaB 6 . X-ray diffraction has con®rmed the cubic hexaboride structure in all cases. Figure 1 shows the electrical resistivity data obtained between 5 K and 300 K from single crystals of Ca 1-x La x B 6 . Already for x 0:005, we ®nd a change to a metallic-like temperature dependence and a drop in resistivity by a factor of ,50 at low temperature, relative to the pure host (x 0). A single band interpretation of the Hall constant obtained using the van der Pauw geometry for the x 0:005 sample indicates electron-like carriers, with a density of 0.005 electrons per formula unit, as one naively expects for the trivalent La substitution. Our preliminary de Haas-van Alphen experiment on x 0:005 La-doped CaB 6 along (100) reveals two orbits. Interpreting these as orthogonal extremal orbits of an ellipsoid gives an occupied volume (taking account of the ellipsoids at the equivalent X-points in the Brillouin zone) enclosing ,0.01 electrons for unpolarized electrons. This value is within experimental error of 0.005 electrons, given the small number of oscillations observed; the observation of two rather than four extremal orbits is consistent with the small hole pocket being ®lled, leaving only a small ®lling of the conduction band.
The unusual aspect of these La-doped borides is seen in Fig. 2, where we plot magnetization versus ®eld at 5 K for a range of values of x, measured on single crystals of Ca 1-x La x B 6 . A weak ferromagnetic moment is evident, its magnitude peaking near x 0:005 at 0.07 m B per La, with no moment found in crystals with composition Ca 0.80 La 0.20 B 6 (Fig. 3); hysteresis loops for the x 0:005 material are shown in Fig. 3 inset. We ®nd nearly identical magnetic effects with La-doping of CaB 6 , SrB 6 and BaB 6 . In all cases, the maximum moment is found at x 0:005. That this moment is a function of the carrier concentration is supported by data for Ca 0.9975 Th 0.0025 B 6 and Ca 0.995 Th 0.005 B 6 ( Fig. 2 inset): here the moment is maximum at x 0:0025. Because Th is incorporated in a tetravalent con®guration, it will contribute one more electron than trivalent La, and hence if the moment is a function of carrier count, compounds with Th 0.0025 and La 0.005 should have the same moment. In Fig. 4, we show the temperature dependence of the magnetic moment of Ca 0.995 La 0.005 B 6 in a ®xed ®eld of 0.1 T as a function of temperature: the data show loss of magnetization near the Curie temperature, T C , of approximately 600 K.
The essential point is to determine that the observed weak ferromagnetism is an intrinsic effect and not of some extrinsic origin. The magnitude of the ordered moment is ,1 e.m.u. per mol hexaboride for x 0:005 in CaB 6 , and is ,2 e.m.u. per mol for Sr 0.995 La 0.005 B 6 . We ®nd consistently that the moment per mole is at a maximum near x 0:005 in both SrB 6 and CaB 6 . The same systematic study has not yet been made for BaB 6 . We ®nd similarly that Ce-and Sm-doped SrB 6 samples have an ordered moment which peaks at x 0:005. In addition, however, we recognize the expected paramagnetic background due to the local magnetic moment due to the f-electrons of these atoms, which is absent for La-doped material. These data indicate that the doping does not produce the moment through trace rare-earth impurities carried by the high-purity La used for the doping. For the pure alkaline-earth hexaborides, we ®nd that a moment is present sometimes, and that this varies from crystal to crystal. This moment is generally distinctly smaller than that found at x 0:005, usually by at least an order of magnitude. We also ®nd a variation in the temperature dependence of the electrical resistivity of pure SrB 6 and CaB 6 from crystal to crystal; crystals showing weak ferromagnetism have a more metallic temperature dependence of the electrical resistivity. Doping divalent hexaborides with other alkaline earths also produces weak ferromagnetism, for instance in Ca 0.995 Ba 0.005 B 6 . We can understand this variable behaviour of the crystals with no  carrier doping on the basis of the band structure of the alkalineearth hexaborides. Calculations 10 show that the details of the overlap of valence and conduction bands at the X-point of the Brillouin zone depend sensitively on the crystallographic parameter ®xing the location of the borons in the unit cell; this parameter determines the relative length of the inter-and intra-octahedral boron±boron bonds. Even small changes in this parameter in the calculations alter the behaviour from insulating to metallic. So we might expect vacancies and foreign-atom additions to alter signi®cantly the properties of the divalent hexaborides for very small dopings, such as seen in materials with similar band structure features (for example, grey tin and bismuth).
Two suggestions for the origin of the weak ferromagnetism which we have observed are (1) ordered defect moments and (2) ferromagnetic polarization of the low-density electron gas. Because no obvious source for a strong coupling giving rise to a Curie temperature as high as T C 600 K presents itself, the coupling of magnetic moments localized on the La or other unspeci®ed impurities on this scale seems rather implausible. A more likely candidate for the origin of the magnetic polarization emerges from studies of electronic correlations in the low-density electron gas, such as those of Ceperley and Alder 11 . This is a topic of theoretical speculation with a long history, going back to Block and Wigner 1 . The study of Ceperley and Alder is a T 0 K computation, comparing unpolarized and completely polarized states of the electron gas, with ferromagnetism showing up for values of r S of the order 80 a B . (Here r S is the radius of the sphere containing one conduction electron; a B the Bohr radius.) Later calculations have lowered this value to ,20 a B (ref. 12). For x 0:005, we compute r S 15:0 Ê A 28:4 a B , using the Bohr radius for the free electron. A recent calculation by Ortiz, Harris and Ballone 13 treating partially spin-polarized states of the low-density electron gas has, in fact, found evidence that near r S 30 a B the stable state is one with ferromagnetic polarization of the order of 10%; this is essentially our experimental ®nding of an ordered moment of ,0.07 m B per carrier at x 0:005. The natural energy scale here is the Fermi energy, E F ; for free electrons we have E F 0:062 eV 720 K for x 0:005 in CaB 6 , of the order of the observed Curie temperature.
The ferromagnetic ground state of a dilute, three-dimensional electron gas has not previously been reported experimentally. But such a ground state seems to provide a possible description of the weak ferromagnetism reported here, although the temperature scale of the phenomenon is unexpected. Detailed calculations appropriate to the lattice case at ®nite temperature are clearly needed, as is much further experimental elaboration of the details of this ferromagnetism.
M and D. Pines for discussions. Work at NHMFL was supported by the NSF; work at Los Alamos was done under the auspices of the US Department of Energy; and the work at ETH Zu Èrich was supported by the Schweizerische Nationalfonds zur Fo Èrderung der wissenschaftlichen Forschung.
Correspondence and requests for materials should be addressed to Z.F. (e-mail: ®sk@magnet.fsu.edu). Some conjugated polymers have luminescence properties that are potentially useful for applications such as light-emitting diodes, whose performance is ultimately limited by the maximum quantum ef®ciency theoretically attainable for electroluminescence 1,2 .
If the lowest-energy excited states are strongly bound excitons (electron±hole pairs in singlet or triplet spin states), this theoretical upper limit is only 25% of the corresponding quantum