Localized vibrational mode analysis of the resistivity and specific heat of LaB 6 D. Mandrus,* B. C. Sales, and R. Jin Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 Received 22 December 2000; published 14 June 2001 LaB 6 and other hexaborides are inclusion compounds in which the rare earth or other metal ion is weakly bound and sits in an oversized ‘‘cage’’ of boron ions. Here we show that a simple model that treats the La ions as independent harmonic Einstein oscillators embedded in a Debye framework of boron ions successfully accounts for the anomalies in the specific heat and resistivity of LaB 6 . One of the nice features of the model is that the Einstein temperature of the La atoms and the Debye temperature of the boron framework are derived from room-temperature x-ray crystallography data. This feature makes the model easy to apply to other hexaborides and other materials that can be treated as inclusion compounds. The results from this work imply that local modes are likely to be important for understanding the physical properties of all the hexaborides. DOI: 10.1103/PhysRevB.64.012302 PACS numbers: 63.20.Kr, 63.20.Pw, 65.40.Ba The discovery of unexpected ferromagnetism in samples of Ca 1 -x La x B 6 Ref. 1 has brought renewed attention to the anomalous physical properties of CaB 6 , LaB 6 , and hexaborides in general. Although it has long been recognized that hexaborides are cryptoclathrates or inclusion compounds, 2 there has been little effort devoted to under- standing how some of the anomalous physical properties of the hexaborides derive from their clathratelike structure. In this paper we show that neither the resistivity nor the specific heat of LaB 6 can be understood without taking into account the localized vibrations of the La ions. The structure of LaB 6 is illustrated in Fig. 1. In this cubic structure the La is at the center of an oversized ‘‘cage’’ con- sisting of 24 boron atoms. The radius of the sphere encom- passing the La ion is about 3 Å; this can be compared to 1.95 Å atomic radius, the 1.4–1.9 Å metallic radius, and the ap- proximately 1.2 Å ionic radius of La. The La ion, therefore, is weakly bound and can be expected to undergo large ex- cursions from its equilibrium position. The boron frame- work, on the other hand, is rigid and makes the hexaborides hard, with high melting points and low coefficients of ther- mal expansion. Electronically, many properties of LaB 6 can be under- stood from simple electron counting arguments. The boron valence band formed from the 6 boron atoms per unit cell requires a total of 20 electrons per unit cell to fill the bonding orbitals; the 6 boron atoms contribute 18 electrons, and the La ion contributes 3 electrons. Thus, there is effectively 1 conduction electron per unit cell. Hexaborides containing di- valent metal ions are semiconductors or semimetals in accor- dance with this picture. 2 The above structural considerations suggest a simple model of LaB 6 in which the rigid boron framework is treated as a Debye solid and the La ions are treated as independent harmonic oscillators Einstein oscillators. This model is clearly oversimplified in that we expect at least some hybrid- ization between the La and boron modes as has been ob- served in inelastic neutron scattering. 3,4 Nevertheless, this model successfully captures the physics of the situation and has the virtue of great simplicity. A good estimate for the Einstein temperature of the La ions and the Debye temperature of the boron framework can be obtained from room temperature crystallography data, specifically the atomic displacement parameters ADP’s that measure the mean-square displacement amplitudes of an atom about its equilibrium position in a lattice. Crystal- lograpers typically report ADP information as a 3 3 matrix U ij that allows for anisotropic displacements. Often an iso- tropic ADP value U iso is given for each site. U iso corre- sponds to the mean square displacement averaged over all directions and is given by one third of the trace of the diago- nalized U ij matrix. In the case of LaB 6 space group Pm 3 m ), the La ions are in position 1a and cannot have an anisotropic U; the B ions occupy 6f and have two indepen- dent ADP’s: U 11 and U 22 =U 33 . For a monatomic cubic crystal, U iso vs T can be solved exactly within the Debye approximation 5,6 and is given by U iso = 3 h 2 T /  4  2 mk B  D 2    D  +0.25 D / T  , 1 where FIG. 1. Structure of LaB 6 . The La ion is weakly bound and sits in a rigid three-dimensional boron ‘‘cage.’’ PHYSICAL REVIEW B, VOLUME 64, 012302 0163-1829/2001/641/0123024/$20.00 ©2001 The American Physical Society 64 012302-1