A phenomenological description of the pressure sensitivity of the Curie temperature in hole-doped manganites

G. Garbarino¹ and C. Acha²

Laboratorio de Bajas Temperaturas, Departamento de Física, FCEyN, Universidad Nacional de Buenos Aires Pabellón I, Ciudad Universitaria, C1428EHA Buenos Aires, Argentina

¹ Present address: ID27-ESRF - BP 166 cedex 09, 38042 Grenoble, France, EU.

² Also fellow of CONICET of Argentina

E-mail: acha@df.uba.ar

Received 3 June 2009, accepted for publication 26 October 2009
Published 2 December 2009

Abstract. We performed high-pressure experiments in La0.8Ca0.2−xSrxMnO3 (LCSMO)(0 ≤ x ≤ 0.2) ceramic samples in order to analyze the validity of the well-known relation between the A-cation mean ionic radius (rA) and the Curie temperature Tc of hole-doped manganites at a fixed doping level and for doping values below the 0.3 (Mn+4/Mn+3) ratio. By considering our results and collecting others from the literature, we were able to propose a phenomenological relation that considers the systematic dependence of Tc with structural and electronic parameters. The proposed expression predicts fairly well the pressure sensitivity of Tc, its dependence with the A-cation radius disorder and its evolution in the high-pressure range. Considering a Double-Exchange model, modified by polaronic effects, the phenomenological law obtained for Tc can be associated with the product of two terms: the polaronic modified bandwidth and an effective hole doping. PACS numbers: 62.50.-p, 71.30.+h, 75.47.Lx

Introduction

Many efforts have been devoted to determine the relevant electronic and structural parameters that fix the Curie temperature (T_c) of manganites [1–3]. The Double-Exchange model [4, 5] (DE) was initially applied in order to correlate electrical transport properties and magnetic ordering in these compounds. But early experiments such as the temperature dependence of the Hall coefficient [6], the differences in the activation energy between thermopower and conductivity [7], and the isotope effect [8] demonstrated the polaronic nature of the carriers. This evidence showed the necessity for introducing polaronic corrections to the electronic bandwidth that determines T_c.

On the other hand, it was experimentally established that, for hole-doped manganites and particularly for the family A_0.7'A''_0.3MnO₃ (where A' is a trivalent rare-earth ion and A'' a divalent alkaline-earth ion), the resulting A mean ionic radius (r_A) has a clear influence on T_c [9]. Experiments also showed that r_A can be varied both by chemical replacement or by an external pressure [9], where in the former case, both the Mn-O bond distance and the Mn-O-Mn bond angle vary while in the latter case, most of the variation comes from the Mn-O bond distance [2]. Although the T_c(r_A) dependence can be well reproduced by varying the pressure just by considering a linear dependence δr_A = γδP (with γ  3.75 · 10⁻⁴Å/kbar) for P ≤ 20 kbar. In this low pressure range, T_c varies linearly with P, while for higher pressures, T_c reaches a maximum value and decreases for a further increase of P [10, 11]. This behavior seems to be related to pressure-dependent competing interactions, like the ferromagnetic (F) and the antiferromagnetic (AF) coupling between the core spins, as suggested by Sacchetti et al. [12], although other factors that govern the polaronic modified DE model can play a major role. On the other hand, Rivadulla et al. [13] made a good quantitative description of the r_A-dependence of T_c at constant doping based on a mean-field model where the reduction of the volume fraction of the itinerant electrons produced by the phase separation is responsible for the observed behavior. In the same way, the microscopic model introduced by Dey et al. [14] reproduces the changes in T_c observed in manganite films due to bulk and Jahn-Teller strains produced by the substrate mismatch, while Millis et al. [15] points out the extreme sensitivity of T_c to the biaxial strain in respect to the bulk compression contribution.

Empirically, the temperature dependence of the pressure (P) sensitivity of T_c (dlnT_c/dP) was established, which seems to represent a universal behavior for many moderated hole-doped manganites [16]. This curve could be described qualitatively within the small polaron modified DE model, but even so we were far from a convenient quantitative explanation [17]. Even considering polarons in the more suitable intermediate electron-phonon coupling regime did not produce a better understanding [18].

In this paper we present a phenomenological model based on the r_A-dependence of T_c for intermediate-to-large bandwidth AMnO₃ hole-doped perovskites which usefully describes the quantitative dependence of T_c(P). A DE interaction, modified by polaronic effects and also an effective doping of the MnO planes, both controlled by r_A are suggested as the two microscopic ingredients that govern the proposed relation.

Experimental

Previous experiments [9] showed that T_c follows a parabolic dependence with r_A for La_1−yT_yMnO₃ (T = Sr; Ca; Pr) for a fixed doping level y ~ 0.3. Here, in order to test the validity of this dependence for other doping levels (y = 0.2), we performed resistivity measurements as a function of temperature and pressure on La_0.8Ca_0.2−xSr_xMnO₃ (LCSMO)(0 ≤ x ≤ 0.2) ceramic samples. These samples were synthezised following a similar process to the one published elsewhere [19]. The temperature dependence of resistivity was measured using a conventional 4-terminal DC technique in a CuBe pinston-cylinder hydrostatic cell described previously [20]. Pressures up to 10 kbar were applied using a 50% mixture of kerosene and transformer oil as the pressure transmitting medium. Pressure was measured at room temperature by using a calibrated InSb sensor and it remains constant over all the temperature range (within a 10% of variation for a temperature span of 77 K to 350 K) in spite of thermal contractions. Temperature was measured using a calibrated carbon-glass thermometer in good thermal contact with the cell's body.

Results and Discussion

The normalized resistivity as a function of temperature of the LCSMO series with 0 ≤ x ≤ 0.2 and y = 0.2 can be observed in fig. 1. All the curves show a change in the conduction regime that can be associated with a metal to insulator transition (T_MI), which increases with increasing Sr content.

The pressure sensitivity of the resistivity is shown in fig. 2 for samples LCSMO with x = 0; 0.06 and 0.20. Pressure increases both T_MI and the conductivity of these materials.

T_MI does not necessarily coincide with the Curie temperature, so we determine T_c from the resistivity curves as the temperature at which a sudden increase in the logarithmic temperature derivative of the resistivity is observed. It has been shown previously that this coincides with the T_c determined by magnetization measurements [21]. By following this criteria and by calculating the variation on r_A generated by chemical replacements (from Shannon's tables of ionic radii [22]) or by external pressure (assuming that γ is independent of the doping level and that r_A is the only pressure-dependent parameter), we obtain the dependence of T_c as a function of r_A for the LCSMO (y = 0.2) samples, shown in fig. 3.

Figure 3. (Color online) Tc vs. the Sr concentration (x) or the average ionic radius of the cation in the A site (rA) at constant doping y = 0.2. The line is a fit using eq. (1); the fitting parameters are displayed. The inset also shows Tc vs. rA, but where the variation of rA is due to both chemical replacements and external pressure (assuming that δTcM/dP = δβ/dP = 0 and δrA = γδP, with γ = 3.75 · 10−4 Å/kbar).

The data is very well represented by a quadratic law for the whole pressure and doping intervals considered, which indicates that the assumptions we made were quite reasonable. A small departure from the ideal dependence can be observed for r_A  1.227 Å, which coincides with a structural transition reported for this series [23].

From our data and the data already published we can extend the study of the T_c(r_A) dependence for other manganites and for doping levels y in the 0.15 ≤ y ≤ 0.33 range. The obtained T_c(y, r_A) curves, shown in fig. 4, follow the same general behavior: a parabolic law for each doping concentration and a doping-dependent maximum [T_c^M(y)] located at r_A^M  1.255 Å. We propose to represent this dependence by an expression of the form

with y_eff = [1 − β(y)(r_A(x, y) − r_A^M)²], that, as will be discussed later, may represent an effective doping fraction, expressed as a function of r_A, r_A^M and β, an empiric parameter.

Figure 4. (Color online) The phase diagram of T1−yDyMnO3 (where T is a trivalent lanthanide as La, Sm, Nd, and D a divalent alkaline earth as Ca, Sr) for 0.15 ≤ y ≤ 0.33 as a function of rA. Data was extracted from references [9, 23–27]. The dashed lines are parabolic fits corresponding to eq. (1) at constant doping y. The evolution of Tc and rA is also shown for the Sr and Ca-doped LMO samples (i.e. samples with a varying y).

By fitting the data presented in fig. 4 we can determine the doping sensitivity of parameters T_c^M and β, as can be observed in fig. 5. As was shown in a previous study [28], the local structural disorder, generated by the occupation of the A site by cations with different sizes, produces a reduction of the ideal T_c that would be measured in case that this disorder does not exist. The disorder can be quantified by the variance σ² of the A-cation radius distribution. In order to perform the fits, minimizing in this way the contribution of disorder in the obtained parameters, we choose from the T_c(y, r_A, σ) data the points with small σ² (< 10⁻³ Å²).

On the other hand, if we assume that eq. (1) gives the T_c of a manganite with negligible A-cation radius disorder [T_c(x, r_A) = T_c(x, r_A, σ = 0)], we can estimate the T_c of a manganite with a structural disorder σ in r_A as the mean T_c resulting from a uniform distribution of cells with A-cation radii within the interval r_A ± σ. The result gives an expression of the form

which gives a simple explanation of the linear dependence of T_c on σ² already published for the perovskite family A'_0.7A^''_0.3MnO₃ [28]. The σ²'s pre-factor can be calculated from the fitted parameters shown in fig. 5. For y = 0.3 we obtain a value of (17.000 ± 1000) KÅ⁻², quite similar to the experimental data published.

Also, we can qualitatively estimate the influence of disorder on the pressure sensitivity of T_c by taking the pressure derivative of eq. (2). If we compare the data of previous papers [11, 31] it is clear that dσ²/dP is an increasing function of σ, the second term of the right part of the derived equation would indicate a reduction of the expected pressure sensitivity of manganite with increasing σ², as was experimentally obtained previously [32].

Besides, considering the similarity of eq. (1) with the one developed by Bean and Rodbell [33] to describe the coupling of magnetic order to structural distortions, a first order magnetic phase transition at T_c can be predicted for large values of β, as was demonstrated experimentally by Otero-Leal et al. [32].

The expression of the pressure sensitivity of T_c, shown in eq. (3), can be easily obtained from eq. (1) as

By using eq. (3) and the fitted parameters T_c^M(y) and β(y), the pressure sensitivity of T_c at low pressures can be predicted for many compounds. A good accordance between experimental points and the predicted behavior, represented by a shaded area as we considered the doping dependence of the fitted parameters, can be observed in fig. 6. Here, we included data points where T_c was determined by different criteria and techniques (ac susceptibility or resistivity) which accounts for the dispersion of data. We only applied the restriction that the selected data points should be derived from compounds with a small structural disorder in r_A (σ < 3 · 10⁻² Å). Although some of the pressure sensitivities seem to be overestimated, the general behavior is very well predicted as a direct consequence of the validity of eq. (1) and the linear dependence of r_A to describe the general behavior of intermediate-to-large bandwidth manganites in the low pressure range considered here (P < 1 GPa). Indeed, our description based on eq. (1) predicts the observed parabolic evolution of T_c with pressure in the intermediate-pressure range (P < 2 GPa) [10]. Also, if we consider the fact that r_A deviates from a linear behavior [11] and reaches a limit value for P > 6 GPa, we may obtain a simple qualitative explanation of the experimental results where the parabolic T_c(P) law is followed by an asymptotic behavior for increasing pressures [30]. On the other hand, in this high-pressure range, pressure induces additional Jahn-Teller distortions of the MnO₆ octahedra [11], not taken into account by our description, that may be at the origin of the abrupt reduction of T_c reported for some compounds [21, 31].

Figure 6. Pressure sensitivity as a function of Tc for compounds of the La1−yTyMnO3 family (T = Sr; Ca; Y, Dy) in the 0.15 ≤ y ≤ 0.33 range. To guarantee the range of validity of eq. (2) in order to estimate Tc, data points with σ < 3 · 10-3 Å2 were extracted from refs. [9–11, 17, 18, 23, 24, 26, 29, 30] and from our measurements. The shaded area represents the predictions of the model (eq. (3)) taking into account the different values of the fitted parameters (β and TcM) and their uncertainties.

Finally, we would like to gain insight on the physical origin of each term in eq. (1).

As for the DE model T_c ~ Wn, where W is the bandwidth and n is related to the electronic density of itinerant carriers [34], we may associate T_c^M with W, and y_eff with n.

As part of the carriers are localized in the polaronic phase, n does not depend solely on the nominal doping value y. Within this framework, the given expression between brackets for y_eff may represent the dependence with r_A of the effective density fraction of itinerant carriers. As shown in the model derived by Rivadulla et al. [13], the value of n needed to determine the T_c(x) dependence at constant doping (y = 1/3) for the system La_2/3(Ca_1−xSr_x)_1/3MnO₃ should take into account this polaronic reduction of the itinerant carriers (in their notation n = 1 − n_JT, where n_JT is the volume fraction of electrons in the polaronic phase). Indeed, in the microscopic model developed by Dey et al. [14], T_c is calculated considering an electronic occupancy of the relevant orbitals that is 60–70% of the one expected from the nominal hole-doping level.

This implies that, not only the steric factors that govern the hopping energy and the polaronic coupling constant that modifies the bandwidth will affect T_c of the manganite by changing T_c^M, but also, and quantitatively more important, variations on r_A will determine its value by modifying the effective value of n.

In this way, we may associate T_c^M with structural parameters and with the polaronic narrowing of the bandwidth as [35]

where W₀ is the bare bandwidth, w is the tilt angle in the plane of the Mn-O bond, d the Mn-O bond length, E_b the binding energy of polarons and F the appropriate function that accounts for the polaronic bandwidth reduction.

In fig. 7 we have plotted the relative variation of the experimental T_c with r_A for samples of the LSMO family. In the same figure we considered the relative variation of W₀, based on the modification of the steric factors. Again, it is clear that the DE model is far to explain the experimental variation of T_c. If we include the correction of the effective doping proposed in our phenomenological model a much better agreement is obtained (W₀ y_eff). Finally, if we use the expression , only valid for low doping levels as, in this range, we are near the frontier from strong to intermediate electron-phonon coupling [36, 37], we can additionally estimate the polaronic bandwidth reduction of W₀ by using the appropriate constants [38]. The excellent agreement with the measured data obtained (marked with an arrow in fig. 7) indicates that the association of T_c^M with W is a reliable assumption.

Figure 7. (Color online) The relative variation of the bare bandwidth W0, of the bare bandwidth times the relative effective doping (W0 yeff) and of the experimental Tc (from ref. [26]), as a function of the nominal doping y or of rA. The arrow shows the correction added by polaronic effects (only valid for low doping levels).

Conclusions

The empirical law that determines the T_c of T_1−yD_yMnO₃ compounds as a function of r_A and the doping level y was experimentally extended, using our data and data already published in the literature for dopings in the 0.15 ≤ y ≤ 0.33 range. For these compounds and in the pressure range where a linear dependence of r_A with pressure is still valid, the influence of cationic disorder and the pressure sensitivity of T_c was quantitatively described by an empirical relation. This expression can be derived from the DE model, where the T_c^M parameter would be related to the polaronic-modified bandwidth, while a second multiplying term should be included, controlled by r_A and associated with an effective doping level fraction.

Acknowledgments

We would like to acknowledge financial support by CONICET Grant PIP 5609 and UBACyT X198. We acknowledge technical assistance from C. Chiliotte, D. Giménez, E. Pérez Wodtke and D. Rodríguez Melgarejo. We are also indebted with A. G. Leyva for providing the samples and with V. Bekeris for a critical reading of the manuscript.

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