MEASUREMENTS OF THE CeAL 2 AND Lal2
RESISTIVITIES UNDER PRESSURE: VARIATIONS
OF THE KONDO EFFECT AND OF THE
CRYSTALLINE FIELD OF CERIUM

Denis Jerome, M. Nicolas

► To cite this version:

Denis Jerome, M. Nicolas. MEASUREMENTS OF THE CeAL 2 AND Lal2 RESISTIVITIES UNDER PRESSURE: VARIATIONS OF THE KONDO EFFECT AND OF THE CRYSTALLINE FIELD OF CERIUM. Solid State Communications, 1972. [arXiv:04983396v1]

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MEASUREMENTS OF THE CeAl₂ AND LeAl₂ RESISTIVITIES UNDER PRESSURE:
VARIATIONS OF THE KONDO EFFECT AND OF THE CRYSTALLINE FIELD OF CERIUM

M. Nicolas-Francillon, A. Percheron and J.C. Achard

Laboratoire des Terres rares, CNRS, 1 Place A. Briand, 92, Bellevue, France

O. Gorochov

Laboratoire de Chimie Minérale, Faculté de Pharmacie, 4, Avenue de l'Observatoire, Paris 6*, France

B. Comut

Centre de Recherches sur les Très Basses Températures, Cedex 166, 38, Grenoble-Gare, France

and

D. Jerome and B. Coqblin

Physique des Solides*, Bât. 510, Faculté des Sciences, 91, Orsay, France

(Received 12 July 1972 by P.G. de Gennes)

The electrical resistivity of CeAl₂ and LaAl₂ compounds has been measured in the 1.3-300°K temperature range and under pressure up to 15 kbar. The pressure dependence of the magnetic CeAl₂ resistivity has been accounted theoretically by the influence of the crystalline field on the Kondo effect, the latter varying under pressure, because of the decrease of the distance between the 4f level and the Fermi level.

CERIUM impurity diluted in lanthanum or yttrium is magnetic and presents a Kondo effect connected to the large resonant scattering effect which arises from the mixing between conduction and 4f electrons, because the cerium 4f level is close to the Fermi level. Moreover, the negative logarithmic behaviour of the low temperature resistivity of LaCe¹ and YCe² alloys and the variation (dTc/dc) of the superconducting temperature Tc with a cerium concentration c in LaCe alloys³ are strongly varying with pressure. These behaviours have been previously explained within the resonant scattering theory, by considering the increase of the 4f level with respect to the Fermi level when applying pressure.⁴

The purpose of this letter is to discuss new experimental measurements on the resistivity of the CeAl₂ compound under pressure, in order to study its Kondo effect. But moreover, CeAl₂ presents a large crystalline field (CF) effect⁵,⁶ which changes deeply the nature of the Kondo effect with respect to the situation of dilute alloys with cerium impurities.⁷,⁸ So, we will firstly remember the main points of the theoretical model describing such a compound,⁸ then we will present the experiments and finally we will discuss the experiments in the framework of the theoretical model.

Within the theoretical model explained in reference 8, the derived effective exchange Hamiltonian describes the resonant scattering

<footnote>* Laboratoire associé au C.N.R.S.</footnote>
character of cerium and takes into account combined spin and orbit exchange scattering.⁸,⁹

For a cerium atom, the large spin-orbit coupling leads to a ground state of total angular momentum $j = 5/2$, which is then split by the CF. In cubic CeAl₂, the CF effect gives a doublet ground state $\Gamma_7$ at an energy $E_1$ (<0) with respect to the Fermi energy and a quartet excited state $\Gamma_8$ at an energy $E_2$ (<0). Let us call $\Delta = E_2 - E_1$ (>0). If we call $M$ the eigenvalue of a 4f eigenfunction in presence of the CF (for example an eigenfunction of $\Gamma_7$ and $\Gamma_8$ in the cubic CeAl₂ case), $C_{\vec{k},M}$ the corresponding creation operator for a 4f electron and $C_{\vec{k}',M'}$ the corresponding creation operator from a conduction electron partial wave function of wavenumber $k$, the effective Hamiltonian is given by:

$$ H = \sum_{k, M} \epsilon_k n_{k,M} - \sum_{\substack{k', M' \\ \vec{k}, \vec{k}'}} J_{M'M'} C_{\vec{k}',M'}^\dagger C_{k,M} (C_{M,M'}^\dagger C_{M',M}) \\ - \delta_{MM'} \langle n_M \rangle + \sum_{k, k', M} \tilde{\mathcal{O}} C_{\vec{k}',M}^\dagger C_{k,M'} \quad (1) $$

The exchange integral $J_{MM'}$ is given by:

$$ J_{MM'} = \frac{|V_{\vec{k},f}|^2}{2} \left( \frac{1}{E_M} + \frac{1}{E_{M'}} \right) \quad (2) $$

where $\tilde{\mathcal{O}}$, taken here as independent of $M$, gives the direct scattering term of (1) and $V_{\vec{k},f}$ is the mixing parameter between conduction and 4f electrons.

Then, the third-order perturbation-theory resistivity $\rho$ has been computed exactly for the CeAl₂ case of two levels split by the CF.⁸ The typical behaviour of $\rho$ presents a peak at a temperature around $\Delta$ and, at high and low temperatures compared to $\Delta$, $\rho$ behaves proportionally to $\ln T$. The resistivity is a function of $J_{11}$, $J_{12}$ and $J_{22}$, but we can present the results for $\rho$ by taking all the $J_{MM'}$ equal to $J$. Within this simplifying approximation, the two limits of $\rho$ for $kT \ll \Delta$ and $kT > \Delta$ have $\ln T$ behaviours given by:

$$ \rho = R_S^i + 2An(E_F)J^2 \frac{\lambda_i^2 - 1}{(2j + 1)} \ln \frac{kT}{D^i} \quad (3) $$

where the spin disorder resistivity $R_S^i$ is:

$$ R_S^i = A \left[ \tilde{\mathcal{O}}^2 + J^2 \frac{\lambda_i^2 - 1}{\lambda_i (2j + 1)} \right] \quad (4) $$

$A$ is a constant parameter:

$$ A = \frac{3m^2\pi v_0 c}{e^2 \hbar^3 k_F^2} \quad (5) $$

$m$ is the electron mass, $v_0$ the atomic volume, $c$ the cerium concentration, $k_F$ the Fermi wave-vector, $n(E_F)$ the density of states for one spin direction. In the low temperatures limit, $\lambda_i$ is the degeneracy of the ground state which is the only one occupied: $\lambda_i = \lambda_0 = 2$. In the high temperatures limit, $\lambda_i$ is equal to the total degeneracy $\lambda_\infty = 2j + 1$ of the 4f level, because all the states are equally occupied. The ratio of the $kT \ll \Delta$ and $kT > \Delta$ logarithmic slopes of $\rho$ is then equal to 3/35. The parameter $D^i$ is equal to the theoretical cut-off $D/2.4$ for $kT > \Delta$, while it is connected to $D$ and $\Delta$ for $kT \ll \Delta$, as explained in reference 8.

The experimental normal pressure magnetic resistivity $\rho$ of CeAl₂, i.e. the difference between CeAl₂ and LaAl₂ resistivities, goes through a maximum around 4°K due to magnetic ordering, has a minimum around 15°K and a broad maximum around 100°K and at last decreases logarithmically till room temperature.⁶,⁸ A good fit of $\rho$ was previously⁸ found by use of the preceding model, except at low temperatures where magnetic ordering is present. The different parameters have been deduced as follows: $\Delta$ is of order the temperature of the broad maximum of $\rho$, while the $\ln T$ behaviour of $\rho$ for $kT > \Delta$ gives the $J_{MM'}$ values (or the $E_M$ values) and $D$. Finally, the ratio between the $kT \ll \Delta$ and $kT > \Delta$ resistivities leads to the evaluation of $\tilde{\mathcal{O}}$ according to (4).

Since $J_{MM'}$ depends on $E_M$ and $E_{M'}$ according to (2) and since the $|E_M|$ values are expected to decrease with pressure in CeAl₂ as in the case of LaCe or YCe alloys. It was previously suggested to measure the pressure dependence of $\rho$ for CeAl₂, in order to study the Kondo effect and especially to observe the increase of the
<figure><img src="image_2.png" /></figure>

FIG. 1. Experimental plots of the total resistivity of CeAl₂ and LaAl₂ vs. temperature at normal (●), 8 kbar (Δ) and 16 kbar (○) pressures. The residual resistivity (5.3μΩ × cm) of LaAl₂ was subtracted from the figure. The lines are here only to guide the eye and are not theoretical curves.

absolute value of the high temperatures slope of the ρ vs. lnT curve.⁸

The definite compounds CeAl₂ and LeAl₂ were prepared by melting weighed amounts of the components (Al: 99.99%; La: 99.9%; Ce: 99.9%) in sealed tantalum crucibles. Homogenization was achieved by turning over each sample and remelting at least twice and holding them ten days at 950°C. Metallographic examination of etching samples by Keller solutions and X-ray diffraction showed that the samples were single phase and homogeneous. These compounds crystallize in the f.c.c. system MgCu₂ type structure. The lattice parameters, computed by Nelson/Riley extrapolation are: *a* = 8.147 ± 0.002 Å for LeAl₂ and *a* = 8.062 ± 0.002 Å for CeAl₂, in good agreement with those previously reported.¹⁰

The variation of the magnetic susceptibility with temperature for CeAl₂ has been measured between 10 and 300°K and is in agreement with previous results.¹¹ The magnetic moment deduced from the Curie-Weiss law between 100 and 400° is 2.6μB and corresponds to the 4f¹ ionic configuration of cerium. The magnetic susceptibility

of LeAl₂, which is equal to 0.623 × 10⁻⁶ e.m.u/g at room temperature, is almost insensitive to the temperature.

The resistivity measurements are made by the standard four lead continuous current method. The solderings on the samples have been done either by silver paint for CeAl₂ or by indium ultrasonics for LaAl₂. The temperature is given by an As-Ga diode between 4.2 and 300°K and by helium pressure down to 1.3°K. The high pressure apparatus has been described elsewhere.¹²

The resistivity curves of a CeAl₂ and a LaAl₂ sample, measured between 1.3 and 300°K at normal, 8 and 16kbar pressures, are shown on Fig. 1. The resistivity of LaAl₂ was not found to vary experimentally under pressure within the experimental accuracy, while the resistivity of CeAl₂ varies appreciably with pressure. So, this variation is surely due to the magnetic behaviour of cerium atoms. The magnetic resistivities, i.e. the differences between CeAl₂ and LaAl₂ resistivities, are plotted for these three pressures as a function of lnT on Fig. 2.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Plot of the magnetic resistivity of CeAl₂, i.e. the difference between the resistivities of CeAl₂ and LaAl₂, vs. temperature in logarithmic scale. The points give the experimental measurements in normal (●), 8 kbar (Δ) and 16 kbar (○) pressures and the lines give the corresponding theoretical curves with the parameters presented in the text and in Table 1.</figcaption></figure>

<table><caption>Table 1. Values of the parameters used for deducing the theoretical plots of Fig. 2 at different pressures</caption><thead><tr><th>Pressure<br>(in kbar)</th><th>0</th><th>8</th><th>16</th></tr></thead><tbody><tr><td>|E<sub>1</sub>|<br>(in °K)</td><td>650</td><td>630</td><td>605</td></tr><tr><td>|E<sub>2</sub>|<br>(in °K)</td><td>540</td><td>535</td><td>520</td></tr><tr><td>Δ<br>(in °K)</td><td>110</td><td>95</td><td>85</td></tr><tr><td>Θ<br>(in eV)</td><td>0.215</td><td>0.225</td><td>0.24</td></tr><tr><td>D<br>(in °K)</td><td>410</td><td>380</td><td>330</td></tr></tbody></table>

According to the experimental data of Fig. 2, the temperature of the maximum of ρ around 5°K decreases and the slope of the ρ vs. lnT curve below roughly 10°K increases rapidly to absolute value under pressure. These points, which are connected to the magnetic ordering of cerium atoms, cannot be accounted here. So, we have fitted the experimental results by the theoretical model of reference 8, from roughly 10°K to room temperature. We have kept constant with pressure the four parameters m, v₀, n(E<sub>F</sub>) and V<sub>κf</sub> and we have taken the same values as

in reference 8: m = 3 a.u., v₀ = 234 a.u., n(E<sub>F</sub>) = 2.2 states/eV atom, V<sub>κf</sub> = 0.07 eV. On the contrary, we have made the four parameters, E₁, Δ, Θ, and D, vary in order to obtain the best fit to the experiments and especially the broad maximum and the high temperatures lnT behaviour of ρ. The theoretical curves are plotted on Fig. 2 and the corresponding values of the parameters are given in Table 1. From Table 1, the variations of D and Θ under pressure are not physically important, because D is a theoretical parameter and the evaluation of Θ comes essentially from the low temperatures behaviour of ρ and consequently depends on the residual resistivity which is not precisely known. The two physically important deductions coming from Table 1 concern the variations under pressure of |E₁| and Δ. So, let us finally discuss these points:

(1) The absolute value of the ρ vs. curve slope at high temperatures increases by roughly 25 per cent under a 16 kbar pressure, showing unambiguously the unexpected decreases of |E₁| and |E₂| under pressure in CeAl₂: |E₁| decreases by 7 and |E₂| by 4 per cent. However, these variations are much smaller than in LaCe¹ alloys. An explanation of this discrepancy can be probably found in the different compressibilities of La and CeAl₂.
(2) A very interesting effect was also found in the large decrease of $\Delta$ by 25 per cent under a 16 kbar pressure. This measurement, which is, to our knowledge, one of the first experimental determinations of the pressure dependence of $\Delta$, cannot at present be checked theoretically.

Measurements of the CeAl₃ resistivity under pressure are at present in progress. The resistivity of the hexagonal CeAl₃ compound was already determined⁶ at normal pressure and smaller $|E_y|$ values were found in CeAl₃ than in CeAl₂,⁸ to explain the larger experimental effect of CeAl₃.⁶

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La résistivité électrique des composés de CeAl₂ et de LaAl₂ a été mesurée de 1.3 à 300°K sous pression jusqu'à 16 kbar. La dépendence en pression de la résistivité magnétique du CeAl₂ a été expliquée théoriquement par l'influence du champ cristallin sur l'effet Kondo, ce dernier variant avec la pression à cause de la diminution de la distance entre le niveau 4f et le niveau de Fermi.