Magneto-optical evidence of double exchange in a percolating lattice

Substituting $Eu$ by $Ca$ in ferromagnetic $EuB_6$ leads to a percolation limited magnetic ordering. We present and discuss magneto-optical data of the $Eu_{1-x}Ca_{x}B_6$ series, based on measurements of the reflectivity $R(\omega)$ from the far infrared up to the ultraviolet, as a function of temperature and magnetic field. Via the Kramers-Kronig transformation of $R(\omega)$ we extract the complete absorption spectra of samples with different values of $x$. The change of the spectral weight in the Drude component by increasing the magnetic field agrees with a scenario based on the double exchange model, and suggests a crossover from a ferromagnetic metal to a ferromagnetic Anderson insulator upon increasing $Ca$-content at low temperatures.

reflectivity and ARPES measurements.
The intimate relation between magnetization and electronic conductivity also emerged from experiments on the Eu 1−x Ca x B 6 series [12,13,14,15]. The Ca-substitution leads to significant changes of the magnetic and electronic properties. The FM transition temperature decreases with increasing Ca content and stoichiometric CaB 6 exhibits no magnetic order [14,15]. Evidence was also established for a spin-filter effect in the charge transport and dynamics for x = 0.4 [12,13].
The electrical transport properties seem to be governed by percolation-type phenomena across the [14,15]. At about x = 0.7, which coincides with the site-percolation limit in a simple cubic lattice, the long range order disappears and for 0.7 < x < 0.9 a spin-glass type ground state is adopted [15].
A recently published approach to explain the behavior of EuB 6 , as well as offering specific predictions for the electronic properties of the Eu 1−x Ca x B 6 series, is based on a double-exchange scenario [16]. This scenario may be regarded as an effective theory for the Kondo lattice problem in the limit of a very small number of carriers. The reduced itinerant carrier concentration places the Fermi level near a magnetization dependent mobility edge, which emerges in the spectral density because of the disordered spin background and/or Ca-doping. An FM metal to insulator crossover is expected as a function of the position of the Fermi level with respect to the mobility edge, which can be tuned by the Ca-content [16]. The model also addresses [17] the region of stability of magnetic polarons in the paramagnetic phase near T C [18].
The goal of this paper is to present and analyze our magneto-optical data of the Eu 1−x Ca x B 6 series. Replacing Eu by Ca has direct consequences on the electrodynamic response. It influences the distribution of the spectral weight between the metallic (Drude) component and excitations at non-zero energy in the absorption spectrum for different Ca-contents at different temperatures and magnetic fields. Our analysis provides support for the phase diagram that emerges from the double-exchange model predictions [16]. The boundary between the metallic and the insulating ferromagnetic state at low temperatures is found to be close to x = 0.5.
Eu 1−x Ca x B 6 single crystals of high structural quality were prepared by solution growth from Al flux, using the necessary high-purity elements as starting materials [14,15]. The optical reflectivity R(ω) was measured in a broad spectral range from the far infrared (FIR) to the ultraviolet, and as a function of both temperature (1.6-300 K) and magnetic field (0-7 T) [1,3]. Because of the broad coverage of spectral range, R(ω) spectra allowed for a reliable Kramers-Kronig (KK) transformation providing the complete absorption spectrum represented by the real part σ 1 (ω) of the optical conductivity [19,20]. At low frequencies, i.e., below our low frequency experimental limit of about 30 cm −1 , R(ω) was extended using the Hagen-Rubens extrapolation, and inserting the σ dc values obtained with d.c. transport measurements [6,12,14]. Above the highest measurable frequency, R(ω) was extended into the electronic continuum with the standard extrapolations [19,20].
The temperature and magnetic field dependence of R(ω) for EuB 6 was already presented and discussed in Refs. 1 and 3. Here, we complement those results with data for x =0.3, 0.4 (Ref.
13), 0.55 and 0.8. For the presentation in Fig. 1, we chose spectra that were recorded at 10 K (i.e., T > T C ) for all compounds, and in magnetic fields of 0 T and 7 T. R(ω) is progressively enhanced with increasing magnetic field for x = 0.3 and 0.4, while for x = 0.55 and 0.8, basically no field dependence was registered. Although R(ω) exhibits metallic character for all these Ca-contents, the onset of the plasma edge in R(ω) is quite broad in all compounds. This is distinctly different from the previously observed rapid increase of R(ω) with decreasing ω for EuB 6 (Refs. 1 and 3).
In order to analyze the spectra, we apply the phenomenological Lorentz-Drude approach [19,20]. The fit procedure consists in considering a Drude term for the zero-frequency mode excitation due to the itinerant charge carriers, and an appropriate number of Lorentz harmonic oscillators (h.o.) for the absorptions at non-zero energies. The latter components represent phonon modes, electronic interband transitions and localized-state excitations. Although of phenomenological character, this approach allows for the evaluation of the spectral-weight distribution among the various components of the absorption spectrum. For all Ca-substitutions, we consistently reproduce each spectrum for any combination of temperature and magnetic field with the same fit procedure and the same number of fit components. As an example, we show σ 1 (ω) at 10 K and 0 T for x = 0.8 in Fig. 2. The individual components to the fit may readily be identified. Both Figs. 1 and 2 demonstrate that in this way the raw experimental data for R(ω) and the resulting σ 1 (ω) can be reproduced very well. We can thus disentangle the Drude spectral weight corresponding to the squared plasma frequency from the spectral weight associated with the excitations at non zero energies. The total spectral weight encountered in the excitations at higher energies is proportional to the sum of the squared mode strengths [19,20].
Because of the specific model prediction [16] of a FM metal to insulator crossover, our primary aim is to evaluate the change of spectral weight (SW ) of the metallic component of σ 1 (ω) (red area in Fig. 2) at 10 K between 0 and 7 T. This difference is calculated on the basis of our Lorentz-Drude results and is defined as [21]: ∆SW Drude = SW Drude K, 0 T). The magnetic field of 7 T is high enough to drive the system into magnetic saturation for all values of x, such that σ 1 (ω) at 7 T reflects the maximum metallicity reached by increasing the magnetic field. The temperature of 10 K was chosen to be close enough to, but above, T C for each compound [22]. In order to allow for a comparison for different Ca-contents, we renormalize the change of the Drude spectral weight ∆SW Drude by the total spectral weight (red plus yellow areas in Fig. 2  schematically define the interval of T and Ca-content, where a cluster spin glass phase was established [14,15]. with the microscopic mechanisms that are considered in the low-density double-exchange model applied to a percolating lattice [16]. The crucial detail of this description of the interaction between the conduction electrons and the classical (S = 7/2) localized magnetic moments is not the usual strong Hund coupling condition (J H ≫ t), but rather the extremely reduced density of itinerant carriers that permits the high-energy electronic states to be projected out. Part of the experimental evidence [4,5,6,12,13,14,15] for the strong interplay between transport and magnetisation in the hexaborides is then the consequence of Anderson localization effects. In this sense, the magneto-transport in 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 [22] We confirmed that any choice of temperature T C < T < 10 K for each Ca-content leads to equivalent results.
[23] The band curvature also depends on the strength of disorder.
[24] In Ref. 16, the scattering is due to spin disorder and also from the disorder induced by the random position of atoms. The treatment of scattering processes is more involved in the case of the spin glass phase close to percolation. This requires a detailed understanding of the spin glass nature, an issue which is beyond the scope of the theory [16].