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Transport and optical properties of magnetic hexaborides

Vitor M. Pereira, Eduardo V. Castro, João M. B. Lopes dos Santos
Antonio H. Castro Neto (Boston University), L. Degiorgi (ETH, Zurich)

Large magnetoresistance effects are known to occur in Eu-based hexaborides [Phys. Rev. B 61, 4174 (2000)]. As a consequence - and following a series of experiments which unveiled intriguing connections between their magnetic, transport and optical properties - the series of compounds R_(1-x)A_xB_6, where A is an alkaline-earth metal such as Ca or Sr, and R a rare-earth magnetic ion, has recently attracted considerable interest.


EuB_6 is a ferromagnetic metal, with many intriguing properties like its very small carrier density, which increases upon decreasing the temperature, or an electrical resistivity that drops precipitously below T_C. The theoretical understanding of these and other effects in EuB_6 has been characterized by controversies surrounding their underlying microscopic origins.

One of our central objectives, is to show how the Double Exchange mechanism, combined with a reduced carrier density and the inevitable Anderson localization effects, provides a coherent framework for the interpretation of most experimental measurements pertaining to these hexaborides. As it turns out, the extremely low density regime of the DEM has remained much unexplored, mainly because all attentions were upon the opposite limit, suitable for the manganites. This work tries to fill some of those gaps and clarify others.

In this context, a series of theoretical results were derived for the DEM in the regime of low carrier densities. By substituting the hopping and density parameters adequate in the context of EuB_6, a good agreement between theory and experiment followed. We summarize the overall understanding that this DE-based picture provides regarding the microscopic details of these hexaborides, and how several distinctive experimental signatures can be consistently understood within this framework.

The Model and its Results

The series R_{1-x}A_{x}B_{6}, where A is an alkaline-earth metal such as Ca or Sr, and R a rare-earth magnetic ion, constitutes a family of cubic compounds where a divalent lanthanoid occupies the central position on a cube, surrounded by eight B_6$ octahedra at each vertex. Boron atoms make up a rigid cage, held together by covalent bonds between neighboring B atoms. EuB_6$ is a ferromagnetic metal, ordering at T_C = 15 K, and characterized by a quite small effective carrier density, of order of 0.001 per unit cell, at high temperatures.

Magnetism is found to arise from the half-filled 4f shell of Eu, whose localized electrons account for the measured magnetic moment of 7\mu_B per formula unit. The FM transition temperature is reported to decrease with increasing Ca content and the totally substituted compound CaB_6 exhibits no magnetism.

In the references indicated we have proposed a simple model that describes quantitatively the following properties revealed by the experiments done in EuB_6:

  1. A precipitous drop in the dc resistivity just below T_C, with a change by a factor as high as 50 between T_C and the lowest temperatures;
  2. The large negative magnetoresistance observed near T_C$;
  3. An increase in the number of carriers, by a factor of 2-3, upon entering the ordered phase, as evidenced by Hall effect;
  4. A large blue shift of the plasma edge, seen also for T < T_C, both in reflectivity, R(w), and polar Kerr rotation;
  5. A scaling of the plasma frequency with the magnetization.

The enumerated features constitute a definite case of strong coupling of the magnetization to the transport properties. As will be clear below, the effects of chemical doping with non-magnetic Ca are also considered, and the theory explains qualitatively the following experimental findings:

  1. With doping, x, the metallic regime, found in EuB_6 (x=0), evolves to a semiconducting behavior above T_C;
  2. Just below T_C the carrier density increases by at least two orders of magnitude;
  3. The plasma edge is visibly smeared while the corresponding resonance in the polar Kerr rotation is greatly attenuated;
  4. \rho(T,H) and \omega_p display an exponential dependence in the magnetization;
  5. There remains a significant and rapid decrease of \rho(T) just below T_C, albeit by a smaller factor than in the undoped case.

To understand how this comes about in the doped case consider the following. Band structure calculations seem to agree that the conduction band has a strong $5d$ Eu component. Ca doping not only dilutes the magnetic system but also the conducting lattice. In order to model this effect, the hoping parameter $t_{i,j}$ should then be replaced by $t_{i,j}p_{i}p_{j}$, where $p_{i}=1$ if the site i is occupied by a Eu atom and $p_{i}=0$, otherwise. The microscopic problem thus becomes a DE problem in a percolating lattice which, at T=0, reduces to a quantum percolation problem.

Since Ca and Eu are isovalent in hexaborides one does not expect the number of carriers to depend on x. Nevertheless, since carriers are presumed to arise from defects it is difficult to be specific on this issue. The mobility edge, on the other hand, is very sensitive to the Eu->Ca substitution, and should drift toward the band center reflecting the increased disorder. Therefore, the effects should be much more pronounced in E_c than in E_F. In the paramagnetic regime (T>T_C), and as more Ca is introduced, E_c should move past the Fermi energy at some critical doping $x_{MI}^{P}$, after which the mobility gap becomes positive. Naturally, this determines a crossover from the metallic regime to an insulating behavior for T>T_C, exactly as seen in the doped compounds [Phys. Rev. B 61, 4174, 2000; Phys. Rev. B 66, 212410, 2002]. At finite T the mobile carriers arise from thermal activation across the mobility gap. The resistivity should display a semiconducting behavior with T and its dependence on M should be dominated by an exponential factor $\rho(M)\sim\exp(\Delta(M)/T)$. In fact, using $\Delta(M)\approx\Delta_{0}(1-\alpha M)$, as happens in the non-diluted case for either of the $\Delta(M)\lessgtr0$ situations, we find that $\rho(M)\sim\exp(-\zeta M)$ ($\zeta$ is a constant), as seen in the experiments.

Proposed phase diagram.
From V. M. Pereira et al., Phys. Rev. Lett. 93, 147202 (2004)

As T is lowered below T_C and M increases on the percolating cluster, one expects the crossover from metallic to semiconducting behavior to occur at larger values of x; this is illustrated in above figure by the curved/dashed line separating the FM and FI regions. In the vicinity of this line, a sharp metal--semiconducting distinction is not possible on account of thermal effects, resulting possibly in a bad--metal behavior. At T=$, there is a metal--insulator transition occurring at a concentration $x_{MI}\geq x_{MI}^{P}$, which corresponds to the quantum percolation transition for a small number of carriers. Even though we expect x_{MI} to be close to $x_{MI}^{P}$, the possibility of a semiconducting behavior crossing over to metallic at low T (for some $x_{MI}^{P}

Ferromagnetism induced by the \ac{DE} mechanism is expected to persist past x_{MI} as long as the localization length is greater than the lattice spacing. Naturally, the critical concentration, x_c, where T_C~0, should not be higher than p_c = 0.69$, the classical site--percolation threshold for the simple cubic lattice. The values of $x_{MI}^{P}$ and x_{MI} in the figure vary with carrier density and are expected to be sample dependent, since carriers seemingly arise from defects. Actually, annealing experiments can be quite important for the study of the phase diagram.

Another consequence of the \ac{DE} picture is that within this model T_C scales with . Since $\omega_{p}^{2}$ follows the same scaling, we expect that in the series Eu_{1-x}Ca_{x}B_{6} the squared plasma frequency should scale approximately with T_C.

Recent Experimental Developments

The first relevant set of results came from a relatively comprehensive study of the magneto-transport thoughout the Ca-doped series by Wigger et al. [Phys. Rev. B 69, 125118 (2004)]. These experiments revealed that the Eu$\rightarrow$Ca substitution leads to strong percolation-related effects in the characteristics of both the magnetic order and electrical transport.

Magneto-optical (experiment) phase diagram.
From Caimi et al., Phys. Rev. Lett. 96, 016403 (2006)

Another set of experiments undertook the analysis of the optical response in the EuCaB_6 series, and provides a most interesting and revealing study complementary to the magneto-transport results [Phys. Rev. Lett. 96, 016403 (2006)]. Their primary target was the evaluation of the change of spectral weight of the metallic component of the optical conductivity at T>T_C, between 0 and 7 T. The magnetic field of 7~T is high enough to drive the system into magnetic saturation for all values of x, such that the conductivity at 7~T reflects the maximum metallicity reached by increasing the magnetic field. The figure above shows the variation of the normalized Drude spectral weight as a function of x, in comparison with the variation of T_C across the series. The ranges on the vertical axes were chosen such that T_C(x=0) coincides with the renormalized changes of the Drude spectral weight. The change is spectral weight decreases sharply between x=0 and 0.3, reaching zero at approximately 50% Ca content.

The x dependence of $\Delta SW^\text{Drude}$ reveals the reduction of the itinerant charge carriers with increasing doping. The field-induced enhancement of $\Delta SW^\text{Drude}$ as a function of x and its correlation with the evolution of T_C is fully consistent with the microscopic mechanisms that are considered in the low-density DE model applied to a percolating lattice. The figure shows that the spectral weight of the Drude contribution to $\sigma'(\omega)$ is enhanced when the system is driven from a PM metallic state into full polarization, up to x~0.4. The progressive reduction of this enhancement with x reflects the drift of E_C and the consequently weakened metallic conduction. For higher Ca concentrations, the Drude spectral weight is insensitive to the spin polarization, a signal that the mobility edge went past the Fermi energy and thus the polarization no longer releases any of the localized states. The decay to zero in the curves for $\Delta SW^\text{Drude}/SW^\text{Total}$ is equivalent to the T-dependent crossover line in the theoretical phase diagram (dashed line in the first figure). The tail-like behavior of $\Delta SW^\text{Drude}/SW^\text{Total}$ for x>0.4 can be understood as originating from the nonzero temperature excitations of carriers across the mobility gap. It is also significant that the results do not depend on the magnetic field used for the normalization of $\Delta SW^\text{Drude}$, as evidenced by the two curves, where $\Delta SW^\text{Drude}$ is normalized by $SW^\text{Total}$ at either 0 or 7 T.

With respect to magnetic order, theis experimental phase diagram shows that the Curie temperature decreases with x, as expected if site percolation is important. Unlike the Drude spectral weight, long-range magnetic order survives until the Ca concentration coincides with the threshold of site percolation. This is in accord with the DE scenario in which the effective magnetic coupling is the result of the electron itinerancy among sites with localized moments. From the optical point of view, the results in the above perfectly reflect the phase diagram predicted in the first one: up to x_{MI}~0.4 one has a metallic ferromagnet; for higher Ca content the system remains a ferromagnetic Anderson insulator until it reaches the percolation threshold. Near and above the percolation threshold the number of disconnected Eu-rich magnetic clusters becomes significant. Even though the tendency should be towards ferromagnetism, it is not surprising that the regime above x~0.7 (beyond percolation, and at very low temperatures) seems to be characterized by glassy magnetism [Eur. Phys. J. B 46, 231 (2005)], on account of the possible presence of superparamagnetic clusters and competing dipolar interactions at such extreme dilutions of the magnetic moments.

The fact that this optical phase diagram strongly resembles the phase diagram coming from our DE picture, underlines the significance of localization effects in the description of these compounds.


Proposed Phase Diagram
Magneto-Optical Experimental Phase Diagram