High-Pressure Phase Diagram in the Manganites: a Two-site Model Study

The pressure dependence of the Curie temperature $T_C$ in manganites, recently studied over a wide pressure range, is not quantitatively accounted for by the quenching of Jahn-Teller distortions, and suggests the occurrence of a new pressure-activated localizing processes. We present a theoretical calculation of $T_C$ based on a two-site double-exchange model with electron-phonon coupling interaction and direct superexchange between the $% t_{2g}$ core spins. We calculate the pressure dependence of $T_C$ and compare it with the experimental phase diagram. Our results describe the experimental behavior quite well if a pressure-activated enhancement of the antiferromagnetic superexchange interaction is assumed.


I. INTRODUCTION
Rare-earth manganites (A 1−x A ′ x MnO 3 , where A is a trivalent rare earth and A ′ a divalent alkali earth) have been the object of renewed interest owing to the discovery of the Colossal Magneto-Resistance (CMR) 1 exhibited by several of these compounds over the 0.2 ≤ x ≤ 0.5 doping range. The physics underlining the properties of CMR-manganites is very rich and not yet completely understood, 2 although CMR is commonly described in the framework of the Double-Exchange (DE) model. 3 DE qualitatively accounts for the closely related phenomena of CMR and temperature-driven transition from the paramagnetic insulating phase (high temperature) to the ferromagnetic metallic one (low temperature). Nevertheless, the remarkable quantitative disagreement with the experimental data indicates that additional and different physical effects must be introduced. 4 The key role of the Jahn-Teller (JT) effect, i.e., a spontaneous distortion of the MnO 6 octahedron, which reduces the energy of the system by lifting the degeneracy of the e g levels, was pointed out by Millis et al. 4 and it is nowadays widely accepted. This effect is naturally associated with a sizeable electron-phonon (el-ph) coupling, which reduces the electron mobility, leading to the formation of small polarons and competing with the delocalizing DE mechanism. A complete description of manganite properties at microscopic level is further complicated by the presence of an antiferromagnetic (AF) Super-Exchange (SE) interaction between the S = 3/2 spins of the localized t 2g electrons, which is relevant in the electron-doped regime (x > 0.5) and could play a role also in the CMR region.
High pressure is a powerful tool to investigate the role of the JT effect in manganites. The application of an external pressure produces a symmetrization of the MnO 6 octahedra, thus reducing the lattice distortions and enhancing carriers mobility. Therefore, applied pressure causes an increase of the metal-insulator transition temperature T IM , which in these systems is almost coincident with the Curie temperature T C . An almost linear increase of the transition temperature T C with pressure P has been observed in several manganites over a moderate pressure range (0-2 GPa). 5,6 In a recent paper the effect of pressure on the metal-insulator transition temperature in La 0.75 Ca 0.25 MnO 3 (x = 0.25) has been studied up to 11 GPa by means of optical measurements. 7 The results clearly show that the linear dependence of T C (P ) is limited only to the low pressure regime (P < 3 GPa), whereas in the high-pressure regime (P > 6 GPa) the T C (P ) curve bends down and it seems to approach an asymptotic value. This behavior, consistently with previous results from high-pressure spectroscopic measurements, 8,9 has been ascribed to the onset of a localizing, pressure-activated, mechanism competing with the natural pressure-induced charge delocalization.
The present paper aims at explaining the experimental T C (P ) behavior using a simple theoretical approach which enables a deep understanding of the microscopic mechanisms driving the observed pressure behavior. We propose a model which contains, besides the standard DE and el-ph coupling, also AF-SE coupling between the t 2g core spins, which may provide the new localizing mechanism at high-P invoked to account for the experimental behavior. 7,8,9 It has been proposed that the competition between AF-SE and DE accounts for some of the magnetic properties of the manganites. 10 The possible role of the AF-SE term in providing the high-pressure extra localizing channel is, indeed, quite reasonable, since the superexchange between t 2g orbitals may be thought as proportional to the square of some hopping integral t t between the t 2g orbitals of neighboring manganese ions. The hopping integral, arising from the overlap of the atomic wavefunctions is expected to increase with increasing pressure.
An extremely simple, yet efficient, model to compute T C (P ) is the two Mn-site cluster, which represents the minimal model including DE, el-ph coupling, and SE interactions. The simplicity of this model enables to carry out exact calculations at finite temperature, which are fundamental to describe the experimental P − T phase diagram. The two-site model (TSM) here presented has already been applied in Ref. 11, where it has been shown to successfully catch the relevant physics of manganites at T = 0 owing to the extremely short-range nature of the interactions occurring in these systems. Here we extend the study to finite temperature and discuss the dependence on pressure of the various parameters in the Hamiltonian. In this way we treat, at the same level, the quenching of the JT interactions and the role of AF-SE coupling in determining the pressure dependence of the metal-insulator transition. We show that the experimental results are well described if the AF-SE interaction is assumed to be renormalized proportionally to the square of the hopping matrix element.
The paper is organized as follows: in Sec. II we introduce the model and the details of the calculation; In Sec. III the pressure dependence of T C is reported and discussed, while Sec. IV is dedicated to the conclusions.

II. MODEL AND CALCULATIONS
The TSM has been extensively employed in the study of polaronic systems, since it is the simplest model able to describe the crossover from a metal to an almost localized small polaron, which is reflected in both ground state 12,13 and the spectral properties. 14,15 The DE-TSM including el-ph coupling has been already studied at T = 0 11 and its relevance to the case of CMR manganites has been discussed. 11,16,17 In particular, the correct ground state has been obtained in several coupling regimes. The qualitative difference between quantum (S=3/2) and classical (S = ∞) spin cases was also pointed out, showing the importance of quantum fluctuations of the core-spins in a proper study of manganites. 11 In the present paper, we extend the model of Ref. 11 to finite temperature and focus our attention on the temperature-driven crossover from paramagnetic to ferromagnetic ordering occurring in manganites. In such a minimal cluster no phase transition with long-range order can occur but, owing to the very short-range nature of the interactions in manganites, predictions on possible instabilities taking place in thermodynamic limit can be inferred.
The hamiltonian of our TSM reads: In the electronic part (first three terms), t is the hopping integral between the e g levels, c † i,σ (c i,σ ) is the creation (annihilation) operator for an electron of spin σ on site i, J H is the Hund's rule coupling, σ i = c † iα σ αβ c iβ is the spin operator on site i ( σ αβ are the Pauli matrices), S i is the local spin ( S i = 3/2) due to localized t 2g core electrons on site i, and J 1 is the AF-SE coupling. The phonon contribution (last two terms) contains an Holstein coupling and an harmonic term, where g is the electron-phonon coupling, n i = Σ σ c † i,σ c i,σ is the electron number operator at site i, and a † (a) creates (annihilates) an Einstein phonon of frequency ω 0 , coupled to the density difference between the two sites. The above form of the el-ph coupling is equivalent to local phonons coupled to the on-site electron density n i , after eliminating the symmetric phonon mode which couples to the total density. 11 For the sake of simplicity, we replace the JT coupling with the standard Holstein el-ph coupling. It has been shown that, as far as the evaluation of T C is concerned, the Holstein coupling in a single orbital model gives results extremely close to a two-orbital model with JT interactions. 18 The independent parameters of this model are J H /t, J 1 /t, ω 0 /t, and the dimensionless el-ph coupling λ = 2g 2 /ω 0 t. As already mentioned, the extreme simplicity of the model does not require further approximations and allows for an exact solution for arbitrary values of the parameters. As an example, the quantum nature of phonons and core spin can be fully taken into account.
In the present calculation we will consider the case of a single electron on the two sites, for which the Hamiltonian (1) has a very small (64 × 64) electronic Hilbertspace (including the quantum S = 3/2 spins of the t 2g electrons), while the infinite phonon Hilbert space necessarily requires a truncation up to some maximum phonon number n ph . Except from the extreme adiabatic case (ω 0 /t ≪ 1) and/or extreme multiphononic regime (λt/ω 0 ≫ 1), where a really large number of phonons can be excited, convergence of the relevant states can be, in general, achieved for relatively small (<100) n ph . In the non-adiabatic and intermediate coupling regime a complete diagonalization of (1) can be carried out numerically. In the case of manganites, where we can estimate ω 0 /t ≈0.5 and λ ≈ 1 (see below), the convergence for the low energy states is achieved already at n ph =10.
The complete diagonalization of H, i.e., the determination of all eigenvalues E n and eigenvectors |ψ n , enables the calculation of the thermal average of the nearestneighbor spin correlation operator µ = ( S 1 · S 2 )/S 2 as This quantity measures the short-range magnetic correlations and it is positive for ferromagnetic phases, negative for antiferromagnetic, and vanishes in the hightemperature paramagnetic phase. As we will show in the next section, a simple analysis of the temperature dependence of the spin-correlation µ(T ), allows us to give a reliable estimate of T C . The choice of this "short-range" estimator of T C is, in our opinion, preferable to thermodynamic estimates in our TSM. The reliability of the TSM in the case of manganites can be assessed also through the analogy with a completely different theoretical approach, the Dynamical Mean Field Theory (DMFT), 19 which has been applied to a lattice model for the manganites by Millis and coworkers. 20 The DMFT is a powerful nonperturbative approach which freezes spatial fluctuations, but fully retains the local (single-site) quantum dynamics, and becomes exact in the limit of infinite coordination. 19 The DMFT maps the original lattice model in the thermodynamic limit onto a self-consistent impurity model which interacts with a quantum bath. As already discussed in Ref. 11, we emphasize the conceptual analogy of these two approaches. In the TS model, the quantum nature of the problem is also completely retained, and each of the two sites "feels" the presence of the other site similarly to the way the impurity site feels the bath within DMFT. We stress that, within the DMFT approach, the system is in the thermodynamic limit. Both the methods, despite their differences, are thus expected to well describe the physics of short-range correlations in manganites. The choice of the TSM allows us to easily include the nearest-neighbors SE-AF interaction which is instead unaccessible by the single-site DMFT and requires an Extended-DMFT study, where the dynamical spin correlation function is also self-consistently evaluated, 19 or a cluster-DMFT approach, which retains short-range dynamical correlations, but substantially increases the computational effort. 21

III. PRESSURE DEPENDENCE OF TC
In order to carry out the comparison between the experimental results reported in Ref. 7 and the T C (P ) values which can be determined using the TSM, we need to know the pressure dependence of the input model parameters t(P ), J H (P ), ω 0 (P ), λ(P ), and J 1 (P ) for La 0.75 Ca 0.25 MnO 3 at least over the 0-11 GPa pressure range. In Ref. 22 the hopping integral t and the Hund's coupling J H for La 0.67 Ca 0.33 MnO 3 (x = 0.33) have been derived from first principles calculations as a function of a/a 0 , where a is the cubic lattice parameter and a 0 its ambient pressure value. It is reasonable to assume the same dependence of t upon a in the x = 0.33 and x = 0.25 materials up to an overall factor, namely t x=0.25 (a/a 0 ) = Ct x=0.33 (a/a 0 ). The proportionality factor C can be evaluated exploiting the proportionality between t and T C at P = 0 [Ref. 2], i.e. through the relation t x=0.25 /t x=0.33 = T C (x = 0.25)/T C (x = 0.33). The pressure dependence of the Hund's coupling J H can be safely assumed to be identical for x = 0.25 and x = 0.33 compounds, since J H is much larger than the other energy scales, and the results of the calculation are therefore only very weakly dependent on its variations. Finally, using the lattice parameter a(P ) recently measured over the 0-15 GPa pressure range for La 0.75 Ca 0.25 MnO 3 , 23 the a/a 0 dependence can be simply converted into a pressure dependence. The increase of the hopping integral t(P ) as a function of pressure is plotted in Fig. 1 in the relevant pressure range (about 12 % at 12 GPa). The pressure dependence of the JT phonon frequency ω 0 (P ), directly obtained from Raman measurements carried out over the 0-14 GPa range, 8 is also useful to get an estimate for λ(P ). For a JT mode, it is reasonable to assume that the electron-phonon coupling g is given by g = √ 2M ω 0 dt(a)/da, where M is the ionic mass. Using the above set of pressure dependent parameters, the estimate for the pressure dependence of the el-ph coupling λ = 2g 2 /ω 0 t can be obtained. The resulting λ(P ) is plotted in Fig 1. The zero-pressure value thus obtained, λ(0) = 1.04, can be checked against an independent estimate of g. The typical Jahn-Teller distortion x 0 = x = 1/ √ 2M ω 0 (a + a † ) can be estimated, in the polaronic regime, as x 0 = 2/M ω 0 g/ω 0 . Using the value x 0 = 0.09Å for La 0.75 Ca 0.25 MnO 3 at P = 0 [Ref. 24], one has λ(0) = 0.94, which compares well with the previous estimate. The remarkable pressure-induced reduction of λ(P ) (about 50% at 12 GPa), is consistent with the observed enhancement of the metallic character of the system and with the increase of the transition temperature T C . 7,8 The last ingredient is the AF-SE coupling J 1 with its pressure dependence. To our knowledge, no direct estimate of this parameter, either experimental or theoretical, is available. An estimate of the zero-pressure value of J 1 (P ) can be obtained by simply imposing the calculated T C (0) be equal to the known experimental value of 220 K for La 0.75 Ca 0.25 MnO 3 at zero pressure. 2 In Fig. 2 we show µ(T ) calculated in the TSM using the known zero pressure values of the parameters (t(0) ≈ 0.20 eV, ω 0 (0)/t(0) = 0.39, J H (0)/t(0) = 12.8, λ(0) = 1.04) and a value of J 1 (0) = 0.047t which is consistent with the experimental value of T C . The crossover from a ferromagnetic state at low temperature, with a large positive value of µ, to a paramagnetic state at high temperature with µ → 0 is evident. The absence of an abrupt variation of µ(T ) is expected since the real phase transition occurring in the thermodynamic limit cannot take place in a finite system and it is replaced by a smooth crossover. In the TSM the crossover is quite broad and the estimate of the critical temperature is not completely straightforward and implies some degree of arbitrariness. For this reason, we propose, and compare, two different methods for evaluating T C . The first one, the halfmaximum method, simply defines the T C as the temperature for which µ(T C ) = [µ(0) − µ(∞)]/2 = µ(0)/2, i.e., for which the spin-correlation becomes one half of its zero-temperature value. The second method is based on a linear extrapolation around the inflection point of the µ(T ) curve. These two methods are schematically represented in Fig. 2 from which it is apparent that the latter method provides a higher value for T C . Interestingly, the two estimates of T C are actually proportional over a wide range of parameter values. Therefore, the choice of one method basically affects only the absolute value of T C and not the internal comparison between results obtained with a given estimator. In the following, we use the halfmaximum estimate to determine the effect of pressure on T C . It is important to note that the precise J 1 (0) value is influenced by the method chosen for determining T C . For example, the determination of T C with the halfmaximum method leads to J 1 (0) = 0.047t(0) whereas the linear extrapolation provides J 1 (0) ≈ 0.060t(0). The slight difference between the zero pressure values is not important since we are mostly interested in the pressure dependence of the parameters rather than in their absolute values. As mentioned before, since the pressure dependence of J 1 is unknown, we have carried out calculations of T C (P ) for three different pressure-dependences of J 1 (P ) namely: i) J 1 (P ) = const.; ii) J 1 (P ) ∝ t(P ); iii) J 1 (P ) ∝ t 2 (P ). Bearing in mind that t(P ) is an increasing function of the pressure (see Fig. 1), the three choices correspond to reduce, keep constant, and increase the ratio J 1 /t upon increasing pressure. The results of the TSM calculations together with the experimental data from Ref. 7 are shown in Fig. 3. Before discussing the accuracy of our results with respect to the experiment, we compare with previous DMFT results of Ref. 20, in which a DE term with classical core spins, a JT-type el-ph interaction with classical phonons and no AF-SE term have been used. By combining the T C (λ)/t values reported for n = 0.75 (i.e., x = 0.25) 20 with the present estimate for λ(P ) and t(P ), the T C (P ) are readily obtained. The T C (P ) values, also shown in Fig. 3, have been divided by a factor 1.1 in order to force the agreement with experimental data at zero pressure. Such a small difference for the T C (0) value confirms the reliability of the present model parameter estimates. Comparing the theoretical results shown in Fig. 3, it is interesting to note that DMFT and the TSM with J 1 ∝ t(P ) provide extremely close results. Beside the previously discussed relationship between the two approaches, this agreement can be explained as follows: The pressure dependence in the DMFT approach is basically determined by λ, while in our TSM λ, ω 0 , and J 1 depend on P . Since ω 0 /t is only weakly dependent on pressure, the choice J 1 ∝ t leads to a TSM in which the pressure dependence is substantially determined only by λ as in the DMFT estimate. This argument explains the observed equivalence between the two approaches and, above all, it points out how the pressure dependence of the el-ph interaction alone does not provide a satisfactory description of the observed experimental high-pressure behavior.
The comparison between our theoretical estimates and experimental data is reported in Fig. 3. The pressure dependence of the AF interaction remarkably affects the calculated P − T phase diagram. For small pressures, the cases i) [J 1 = const] and ii) [J 1 ∝ t] nicely follow the experimental T C , while case iii) [J 1 ∝ t 2 ] gives a lower critical temperature. By increasing the pressure, in the two first cases T C increases too rapidly with pressure, while the third estimate closely follows the experimental data up to 11 GPa. In particular, J 1 = const produces an almost linear behavior of T C (P ) over the whole pressure range. The ability of the J 1 ∝ t 2 data to reproduce the actual experimental behavior is really encouraging, since the t 2 dependence may be seen as the most "physical" approximation for J 1 . The basic hypothesis is that the pressure dependence of the overlap between the t 2g orbitals is similar to the dependence of the same overlap for the e g orbitals, i.e., that for each pressure t t (P ) ∝ t(P ). Under this assumption, the superexchange between core spins J 1 ∝ t 2 t /U (where U is an effective repulsion energy depending on J H , the Mn-O charge transfer energy, and the on-site Coulomb repulsion) becomes in turn proportional to t 2 /U , t being the hopping integral between the e g orbitals. These findings clearly show that the effects of pressure are not only limited to a reduction of the el-ph coupling but a remarkable pressure dependence of the AF interaction has to be introduced to account for the experimental behavior. The same information can be obtained by extracting an "experimental" dependence of the antiferromagnetic term on pressure J 1 (P ). This simply amounts to determine, for each pressure value, the value of J 1 which provides the experimental T C , given that all the other parameters are fixed by the above mentioned conditions.
In Fig. 4, we show J 1 (P ), normalized to the zeropressure value together with t 2 (P )/t 2 (0). The two quantities compare quite well, strengthening the J 1 (P ) ∝ t 2 (P ) ansatz. For sake of comparison the quantity t(P )/t(0) corresponding to the case ii) is also shown in Fig. 4. The above results strengthen the hypothesis that the saturation of T C (P ) observed above 6-7 GPa 7 can be ascribed to the onset of a regime in which the AF interaction is no more negligible and competes with the DE. This hypothesis is supported by several recent highpressure experiments. The occurrence of AF interactions in a DE system leads to charge localization, which in turn may favor the coherence of the JT distortions. This picture is consistent with high-pressure X-ray diffraction experiments, in which the onset of a coherent JT distortion has been observed at about 7 GPa in La 0.75 Ca 0.25 MnO 3 and other compounds of the La 1−x Ca x MnO 3 series. 23,25 It is also worth to notice that the onset of the AF phase in LaMnO 3 is accompanied by a remarkable softening of the B 2g -JT phonon. 26,27 Even if in our case no real AF ordering occurs, the enhancement of the AF interaction suggested by our analysis is consistent with the observed saturation of the same phonon in La 0.75 Ca 0.25 MnO 3 at about 7 GPa. 8 The J 1 (P ) ∝ t 2 (P ) dependence suggests that, upon increasing the pressure, the system evolves from a regime dominated by DE, where the bond compression leads to an increase of T C , to an intermediate regime, where T C is almost constant and independent on pressure, owing to the competition between SE and DE, and eventually to a very high pressure regime dominated by the SE contribution where T C starts decreasing. This picture is absolutely consistent with the pressure dependence of T C (P ) recently observed in several manganites where the three regimes are apparent, 28,29,30 as well as with the data of Ref. 7 for La 0.75 Ca 0.25 MnO 3 , where the system only reaches the intermediate regime for the maximum pressure. The extent of the pressure range over which T C is almost pressure independent is different for the samples studied in Refs. 28,29,30, although the onset of saturation takes place around 4 GPa for all these samples. This pressure value is lower than that observed in La 0.75 Ca 0.25 MnO 3 7 and the difference can be ascribed to the presence of strong coupling and/or high cation disorder which have been shown to have strong effects on the pressure dependence of T C . 7

IV. CONCLUSION
We studied the evolution of the transition temperature of La 0.75 Ca 0.25 MnO 3 with pressure by exactly solving a two Mn-site model which contains all the relevant microscopic interactions acting in the manganites. The theoretical results have been compared with the phase diagram recently measured over the 0-11 GPa pressure range. 7 As input parameters for the model we used pressure dependent data available in literature, except for the direct AF-SE between the t 2g spins, for which, in absence of experimental estimates, we compared three different pressure dependences. We have shown that the pressure dependence law for J 1 is indeed crucial to ob-tain a good agreement with the experiment. Neglecting the effect of pressure on the AF-SE term leads in fact to a sizeable overestimate of the Curie temperature for P > 6 GPa. On the other hand, the theoretical results follow the experimental data over the whole pressure range for J 1 (P ) ∝ t 2 (P ), while the calculation largely overestimates the critical temperature at large pressures if a weaker pressure dependence of J 1 is assumed.
The present results give a clear indication that the observed high pressure behavior of T C can be ascribed to a pressure-induced enhancement of the AF-SE which en-ters in competition with the DE above 6-7 GPa. The idea of a pressure activated AF interaction is, at least, compatible with the previous experimental evidences of the onset of cooperative JT effect 23,25 and phonon softening 8 in the very high pressure regime.
We gratefully thank S. Ciuchi for helpful discussions. M.C. acknowledges hospitality and financial support of the Physics Department of the University of Rome "La Sapienza", and of the Istituto Nazionale per la Fisica della Materia (INFM), as well as financial support of Miur Cofin 2003.