Open Questions in CMR Manganites, Relevance of Clustered States, and Analogies with other Compounds

This is an informal paper that contains a list of ``things we know'' and ``things we do not know'' in manganites. It is adapted from the conclusions chapter of a recent book by the author, {\it Nanoscale Phase Separation and Colossal Magnetoresistance. The Physics of Manganites and Related Compounds}, Springer-Verlag, Berlin, November 2002. The main new result of recent manganite investigations is the discovery of tendencies toward inhomogeneous states, both in experiments and in simulations of models. The colossal magnetoresistance effect appears to be closely linked to these mixed-phase tendencies, although considerably more work is needed to fully confirm these ideas. The paper also includes information on cuprates, diluted magnetic semiconductors, relaxor ferroelectrics, cobaltites, and organic and heavy fermion superconductors. These materials potentially share some common phenomenology with the manganites, such as a temperature scale $T^*$ above the ordering temperature where anomalous behavior starts. Many of these materials also present low-temperature phase competition. The possibility of colossal-like effects in compounds that do not involve ferromagnets is briefly discussed. Overall, it is concluded that inhomogeneous ``clustered'' states should be considered a new paradigm in condensed matter physics, since their presence appears to be far more common than previously anticipated.


I. INTRODUCTION
In this paper -not intended for publication in a formal journal -I present a list of "things we know" and "things we do not know" about manganites and related compounds. This text is adapted from the last chapter of a book I recently finished on manganites [1], after receiving the suggestion by some colleagues of making the "open questions" discussion available to a wider readership. The presentation is very informal to keep the discussion fluid, and it is intended for researchers with some background in manganites (other materials, such as cuprates, are also addressed). Only a handful of references are included here for simplicity. However, close to 1,000 citations can be found in the original source [1] as well as the detailed justification of many of the matter-of-fact statements expressed here, particularly in the "things we know" section. Many reviews are also available [2] with plenty of references: this paper is not a review article but an informal discussion, with many comments and sketchy ideas. Also, some items reflect the personal opinion of the author and may be debatable. In addition, this paper discusses results for other families of compounds which exhibit similarities with manganites. It is conceivable that the knowledge accumulated in Mn-oxides may be applicable to the famous high-T c cuprates, as well as other materials discussed here and in other chapters of [1]. The stability of "clustered" states -often called phase-separated or mixed-phase states -appears to be an intriguing property of a variety of compounds, and "colossal effects" should be a phenomenon far more common than previously believed. There is plenty of work ahead, and surprises waiting to be unveiled. T * where clusters start forming well above the Curie temperature. Independent studies by Burgy et al. and Salamon et al. have characterized this critical temperature as a Griffiths temperature. The Griffiths effects appear to be larger than usual due to phase competition. Thick (thin) lines denote first (second) order transitions. Shown is a tricritical case, but it could be bicritical or tetracritical as well. g is some parameter needed to change from one phase to the other. (Middle) With increasing disorder the temperature range of first-order transitions separating the ordered states is reduced, and eventually for a fine-tuned value of the disorder the resulting phase diagram contains a quantum critical point. In this context this should be a rare occurrence. (Right) In the limit of substantial disorder, a window opens between the ordered phases. The state in between has glassy characteristics and it is composed of coexisting clusters of both phases. The size of the coexisting islands can be regulated by disorder and by the proximity to the original first-order transition. For more details see [1,2,6]. The T * discussed in the previous item -remnant of the clean-limit transition -is also shown.

III. CAN THEORIES THAT DO NOT ADDRESS PHASE SEPARATION WORK TO UNDERSTAND THE CMR EFFECT?
The enormous experimental effort on Mn-oxides has already provided sufficient results to decide whether or not some of the theories proposed in recent years realistically explain the unusual magnetotransport properties of these compounds. Indeed, some of the early proposals for theories of CMR manganites have already been shown to be incomplete and the current leading effort centers around inhomogeneities and first-order metal-insulator transitions. The detail is the following: FIG. 2: Schematic representation of theories for manganites. (a) is a simple "double exchange" scenario, without phase competition. (b) relies on Anderson localization as the origin of the insulating state that competes with ferromagnetism. (c) is based on a gas of polarons above the Curie temperature TC, also without phase competition. In (d), a phase-separated state above the ordering temperatures is sketched. Details can be found in the text.

1.
In theories sometimes referred to as "double exchange", electron hopping above the Curie temperature T C is simply described by a renormalized hopping "t cosθ/2 ". These theories are based on the movement of electrons in a disordered spinlocalized background (see Fig.2 (a), for a crude sketch), without invoking other phases. However, quantitative investigations have shown that this approach does not appear to be sufficient to produce neither an insulating state nor a CMR phenomenon, although this simple idea may be suitable for large bandwidth manganites, such as x=0.4 LaSrMnO.
2 Some theories rely on Anderson localization to explain the insulating state above T C . However, the amount of disorder needed to achieve localization at the densities of relevance is very large (at least in a simple three-dimensional Anderson model). Computational studies [7] locate the critical value at W c ∼16 (in units of the hopping), assuming a uniform box distribution of random on-site energies [-W/2,+W/2], and with the Fermi energy at the band center. Perhaps this large disorder strength crudely mimics the influence of large electron-phonon couplings, strong Coulomb correlations, nanoclusters, strain, and quenched disorder present in the real materials. But, even in this case, it is difficult to explain the density-of-states pseudogap found in photoemission experiments by Dessau's group. Anderson localization does not produce such a pseudogap ( Fig.2  (b)). In addition, experimentally it is clear that the CMR originates in the competition between phases, typically FM-metallic and AF/CO-insulating, but this fundamental effect is not included in simple Anderson localization scenarios, where ordering and phase competition are absent. For these reasons, I believe Anderson localization does not seem to be the best approach to explain manganite physics.
3. Some theories are based on a picture in which the paramagnetic insulating state is made out of a gas of small and heavy polarons (Fig.2 (c)). These theories do not address neutron experiments reporting CE charge-ordered clusters above T C , correlated to the resistivity, nor the many indications of inhomogeneities. A polaron gas may be a good description at much higher temperatures, well above room temperature, but such a state does not appear sufficient in the region for CMR, close to the Curie temperature. The charge-ordered small clusters found experimentally above T C have properties corresponding to phases that are stable at low temperatures, such as the CE-phase. These complex clusters certainly cannot be considered to be mere polarons. They are more like "correlated polarons", as some researchers in this field prefer to call the charge-ordered islands.
Theories based on microscopic phase separation ( Fig.2 (d)) appear to provide a more realistic starting point to manganites since they are compatible with dozens of experiments. However, as shown below, there is plenty of room for improvement as well!

A. Potentially Important Experiments (in random order)
• The evidence for charge-ordered nanoclusters above T C should be further confirmed. It is important that a variety of techniques reach the same conclusions regarding the presence of nanoclusters above T C , definitely ruling out an homogeneous state as the cause of the CMR effect. For instance, it would be important to find out the role played by nanoclusters on optical conductivity results, which thus far has been described mainly as consisting of "small polarons" (Noh and collaborators). How do the nanoclusters manifest themselves in the optical spectra? Recent results at x>0.5 by Noh's group much contribute to this issue, since a pseudogap was observed compatible with photoemission results.
• The existence of the predicted new temperature scale T * above the Curie temperature should be further investigated.
(i) Thermal expansion, magnetic susceptibility, X-rays, neutron scattering, and other techniques have already provided results supporting the existence of a new scale T * , where clusters start forming upon cooling. In fact, very early in manganite investigations, the group of Ibarra at Zaragoza reported the existence of such a scale, in agreement with more recent theoretical and experimental developments. This scale should manifest itself even in the d.c. resistivity, as it does in the high temperature superconductors at the analog T * "pseudogap temperature". Are there anomalies in ρ dc vs. temperature in Mn-oxides as well? Are the T * scales in cuprates and manganites indicative of similar physics? (ii) In addition, the specific heat should systematically show the existence of structure at T * due to the development of short-range order (at T * , even a glassy phase transition may occur, as recently proposed by Argyriou et al.). (iii) The dependence of T * with doping and tolerance factors should be analyzed systematically. Theoretical studies [6] suggest that the tolerance factor may not change T * substantially, although it affects the ordering temperatures dramatically. Is there experimental support for this prediction? (iv) Is the crude picture of the state between T C and T * shown in Fig.3 qualitatively correct?
• X-rays and neutron-scattering studies of (La 1−y Pr y ) 1−x Ca x MnO 3 (LPCMO) are needed to analyze the evolution of charge-ordered nanoclusters. There is considerable evidence that LCMO x=0.3 at temperatures above T C presents charge ordered nanoclusters, correlated with the behavior of the resistivity, as already discussed. It is important to track the intensity and location in momentum space of the peaks associated with charge ordering as La is replaced by Pr. This replacement enhances charge ordering tendencies, as discussed by Cheong and collaborators. It is also important to notice that the substitution of Ca by Sr decreases that tendency. Is there a smooth evolution from LCMO to the regime found by Cheong's group in LPCMO with phase separation at low temperatures, as well as from LCMO to the more "double exchange" homogeneous regime of LSMO?
• Are the mesoscopic clusters found in LPCMO using electron diffraction representative of the bulk? Are there other compounds with the same behavior? The evidence for nanocluster formation appears robust in manganites. However, the evidence for larger structures of the mesoscopic kind is based on a smaller number of experiments: electron diffraction and STM techniques (see Sec. II). Are these results representative of single-crystal behavior? To what extent should one consider two kinds of phase separation, i.e., nanoscale and microscale? Can one evolve smoothly from one to the other as the Curie temperature decreases?
• What is the nature of the ferromagnetic-insulating phases that appear in some phase diagrams? Is this phase truly qualitatively different from the ferromagnetic-metallic phase, or are they very similar microscopically? In other words, is there a spontaneously broken symmetry in the ferro-insulator state? Does charge ordering and/or orbital ordering exist there? Theoretical studies by Yunoki, Hotta, and others have shown the presence of many novel spin ferromagnetic phases, with or without charge and orbital order [8]. There is no reason why these phases could not be stabilized in experiments. In fact, a new "E-phase" of manganites at x=0 has recently been discussed [3]. This is an exciting new area of investigations and plenty of phases found in simulations could be observed experimentally, as recently exemplified by the ferromagnetic charge-ordered phase discussed by Loudon and collaborators (see [3]), which had been predicted by Yunoki et al. [8].
• Atomic resolution STM experiments should be performed, at many hole densities. The very recent atomic-resolution STM results for BiCaMnO by Renner et al. are very important for the clarification of the nature of manganites, and for the explicit visual confirmation of phase separation ideas. Extending experiments of this variety to other hole densities, particularly those where the system becomes ferromagnetic metallic at low temperatures, is of much importance. In the area of cuprates, high resolution STM results by Davis, Pan and collaborators have made a tremendous impact, and hopefully similar experiments can be carried out in manganites.
• The regime of temperatures well above room temperature must be carefully explored, even beyond T * . At T >T * , the CO clusters are no longer formed but a gas of polarons may still be present. Perhaps this leads to another temperature scale, denoted by T pol in Fig.4, where individual polarons start forming? Is the system metallic upon further increasing the temperature? We may have interesting physics even at temperatures close to 1,000 K in this context.
• Is the glassy state in some manganites of the same variety as a standard spin-glass, or does it belong to a new class of "phase-separated glasses"? The issue is subtle since, as of today, proper definitions of glasses and associated transitions are still under much discussion. Can glasses be classified into different groups? Are glassy manganites a new class? What is the actual nature of the "cluster glasses" frequently mentioned in manganites? Important work by Schiffer's group, by Levy, Parisi and collaborators in Buenos Aires, and by many other researchers, has added key information to this subarea of manganite investigations. Studies of time-dependent phenomena in the x=0.5 regime has provided fascinating results, that deserve further research.
Related with the previous item, should nanocluster phase-separated states be considered as a "new state of matter" in any respect? This issue is obviously very important. In order to arrive at an answer, the origin of the nanocluster formation must be fully clarified. Is it purely electronic-driven, or a first-order transition rendered continuous by quenched disorder, or straindriven? If the inevitable disorder related to tolerance factors could be tuned, what happens with the nanoclusters and critical temperatures?
• What is the nature of the charge-ordered states at x<0.5, such as those in PCMO? It is accepted that these states "resemble" the CE-state in their charge distribution, but what is the actual arrangement of spins and orbitals? Is the excess of electronic charge distributed randomly in the x=0.5 CE structure or is it uniformly distributed, for instance by increasing uniformly the amount of charge in the Mn 4+ sites? Theoretical studies are difficult at these hole concentrations. It is important to know whether entropy is large in the CO states at these densities, to justify thermodynamically their existence (as discussed by Khomskii). Otherwise, how can a putative low-entropy CO state be stabilized at high temperature? The opposite, a FM state stable above the CO state, is more reasonable and has been already shown to be the case in simulations by Aliaga and collaborators.
• Is there any compound with a truly spin-canted homogeneous ground state? As far as I know, theoretical studies using robust techniques have not been able to find spin-canted homogeneous states in reasonable models for manganites (of course, if no magnetic field is added). Are there experiments suggesting otherwise? Thus far, experimental evidence for canting can be always alternatively explained through inhomogeneities in the ground state. A counterexample are perhaps bilayers in the direction perpendicular to the planes (see the many results by the Argonne group), but this may be a different kind of state, unrelated to the original proposal by DeGennes that postulated a homogeneous spin-canted state interpolating between FM and AF limits.
• The temperature dependence of the d.c. resistivity of manganites has not been sufficiently analyzed Can non-Fermi-liquid behavior be shown to be present in metallic manganites, as it occurs in many other exotic metals?
• Are the nanoclusters found in manganites and cuprates (see below) characteristic of other oxides as well? The answer seems to be yes. In Chapter 21 of [1], many materials that behave similarly to Mn-and Cu-oxides are listed. Nickelates are another family of compounds that have stripes and charge-ordering competing with antiferromagnetism. Below, other materials with similar characteristics are briefly discussed. These analogies are more than accidents. They suggest that many compounds are intrinsically inhomogeneous. Theories based on homogeneous states appear unrealistic.
• Are Eu-based semiconductors truly described by ferromagnetic polarons as believed until recently, or is a nanocluster picture more appropriate? The recent studies by Lance Cooper's group using Raman scattering suggest the existence of close analogies between Eu semiconductors and manganites. Perhaps phase separation dominates in Eu compounds as well, and the long-held view of Eu-semiconductors as containing simple "ferromagnetic polarons" (one carrier, with a spin polarized cloud around) should be revised.

V. SOME OF THE UNSOLVED THEORETICAL ISSUES
• The study of models for manganites is far from over! Although much progress has been made [1, 2] many important issues are still unexplored or under discussion. For example, the phase diagram in three dimensions of the two-orbitals model may contain many surprises. It is already known that the 1D and 2D models have a rich phase diagram, with a variety of competing phases. In addition, studies by Hotta et al. in two dimensions including cooperative effects have also revealed the presence of stripes at densities x=1/3 and 1/4, and probably others. Then, one can easily imagine a surprisingly rich phase diagram for bilayers or 3D systems. Perhaps the dominant phases will still be the A-type AF at x=0, ferro-metal at x∼0.3, CE-type at x=0.5, C-type at x∼0.75, and G-type at x=1.0. However, the details remain unclear.
• The theory of phase separation should be made more quantitative. Can a rough temperature dependence of the d.c. resistivity be calculated within the percolative scenario? Certainly we already have resistor-network calculations that match the experiments, but not a simple anybody-can-use formula. This is a complicated task due to the difficulty in handling inhomogeneities.
• Is the "small" J AF coupling between localized t 2g spins truly an important coupling for manganites? The Heisenberg coupling between localized spins appears to play a key role in the stabilization of the A-type AF state at x=0. This small coupling selects whether the system is in a FM, A-, C-or G-type state, namely its influence is amplified in the presence of nearly degenerate states. This coupling is also important in the stabilization of the correct spin arrangement for the CE-phase (if J AF is too small, ferromagnetism wins, if too large G-type antiferromagnetism wins), and in the charge stacking of the CE-phase, according to studies by Yunoki, Hotta, Terakura, Khomskii and others. This key role appears not only in Jahn-Teller but also in Coulombic-based theories as well, as shown by Ishihara, Maekawa and collaborators. Are there alternative mechanisms for stabilization of A-, CE-and charge stacking states? The importance of J AF may be magnified by the competition between many phases in manganites, with several nearly degenerate states.
• Why is the CE-state so sensitive to Cr-doping? Experimentally it is not expected that a robust charge-ordered state could be destabilized by a relatively very small percentage of impurities. Can this be reproduced in MC simulations? Note that high-Tc cuprates are also very sensitive to impurities. I believe that materials near percolative transitions are naturally sensitive to disorder. These effects may provide even more evidence for the relevance of inhomogeneous states in oxides.
• For the explanation of CMR, is there a fundamental difference between JT-and Coulomb-based theories? Technically, it is quite hard to handle models where simultaneously the Coulomb Hubbard interactions as well as the electron-phonon couplings are large. However, thus far for CMR phenomena the origin (JT vs. Coulomb) of the competing phases does not appear to be crucial, but the competition itself is. Is this correct?
• In the calculations by Burgy et al. [6], phenomenological models were used for CMR. Can a large MR effect be obtained with more realistic models? Of course the calculations would be very complicated in this context, if unbiased robust many-body techniques are used, due to cluster size limitations.
• Are there models with spin canted homogeneous ground states? Thus far, when models for manganites were seriously studied with unbiased techniques, no homogeneous spin canted states have been identified (at zero magnetic field). Perhaps other models?
• Can a model develop a charge-ordered AF phase at intermediate temperatures while having a ferromagnetic metallic phase at low temperatures? This is quite hard and perhaps can only occur if the CO phase has an associated high entropy, as has been discussed by Khomskii (see similar discussion in the previous section).

VI. ARE THERE TWO TYPES OF CMR?
The manganite (Nd 1−y Sm y ) 1/2 Sr 1/2 MnO 3 investigated by Tokura and collaborators has an interesting behavior, shown in Fig. 5. This compound presents two types of CMR phenomena: (1) At temperatures in the vicinity of 250 K (y=0) and 150 K (y=0.75) a somewhat "standard" CMR is observed. Here, by standard it is understood the CMR behavior described in reviews and [1], with the canonical shape of the resistivity vs. temperature. However, (2) at lower temperatures where the system is insulating, a huge MR effect is observed as well. Results for two values of "y" are shown in the figure. Is this indicative of two independent mechanisms for CMR?
Related with this issue is the recent work of Fernandez-Baca and collaborators studying Pr 0.70 Ca 0.30 MnO 3 , where they reported the discontinuous character of the insulator to metal transition induced by an external field. Those authors argued correctly that the CMR phenomenon can be more complex than the percolation of FM clusters proposed for the standard CMR. Their results are in qualitative agreement with those of Fig.5 where, at low temperatures, a fairly abrupt transition is observed. In fact, it is known that even in percolative processes such as those in LPCMO (see work of Cheong and collaborators), hysteresis has been found in the resistivity. This suggests that a mixture of percolation with first-order features could be at work in manganites, as theoretically discussed by Burgy et al. [1,6].
Can theory explain the presence of two types of CMR transitions? The answer is tentatively yes, although more work is needed to confirm the picture. The main reason for expecting two types of CMR is already contained in the phase diagram of the models studied by Burgy et al. [6] to address the competition of two phases, as schematically reproduced in Fig.6. There, two possible regions with CMRs effects are shown. One corresponds to the "standard" region, at the transition from a "clustered" short-range ordered phase to the FM metallic phase with decreasing temperature (CMR2). However, at low temperatures, and if the quenched disorder is not too strong, the transition between the competing phases can remain of first order in a finite range of temperatures (see Fig.1 and discussion in Sec.II). As a consequence, if the system is very close to the metal-insulator transition and on the insulating side, a small magnetic field can cause a first-order transition between the two phases (in the CMR1 region in the figure), which will imply a dramatic and discontinuous change in the resistivity. This effect has been recently confirmed in simulations of the two-orbital model with Jahn-Teller phonons by Aliaga and collaborators (to be published). Clearly, the theory predicts two types of CMR transitions and this is already in agreement with experiments. Note that if there were an external means to favor the CO/AF phase over the FM phase (for instance by an "external staggered field"), then the process could be reversed and a metal to insulator exotic transition would be induced. See more about this when potential colossal effects are discussed for cuprates.
FIG. 6: Schematic representation of the generic phase diagram in the presence of competing metal and insulator, and for quenched disorder not sufficiently strong to destroy entirely the first-order transition at low temperatures. g is a generic variable needed to transfer the system from one phase to the other. CMR1 and CMR2 are the regions with two types of large MR transitions, as described in the text and in [1].

VII. EVEN MORE GENERAL OPEN QUESTIONS
At the risk of sounding naive, here are very general issues that in my opinion should also be addressed by experts: Can quasi one-dimensional manganites be prepared experimentally? Perhaps chemists may already know that this is impossible, but here let us just say that in the area of cuprates it was quite instructive to synthesize copper-oxides with chain or ladder structures. These quasi-1D systems can usually be studied fairly accurately by theorists, and concrete quantitative predictions can be made, both for static as well as dynamical properties. Regarding the CMR effect, there are already calculations (see Fig.7) that show a very large MR effect in 1D models. In a 1D system a single perfectly antiferromagnetic link is sufficient to block the movement of charge (since its effective hopping in the large Hund coupling limit is zero), creating a huge resistance. Small fields can slightly bend those AF oriented spins, allowing for charge movement and decreasing rapidly the resistivity by several orders of magnitude, as shown in the figure. Should we totally exclude superconductivity as a possibility in manganites? Naively, the presence of superconductivity in transition-metal oxides of the n=3 shell should not be necessarily restricted to just the cuprates. In the Cu-oxide context, superconductivity appears when the insulator is rendered unstable by hole doping. Antiferromagnetic correlations are among the leading candidates to explain the d-wave character of the superconducting state. In the high-doping side of the superconducting region, a Fermi liquid exists even at low temperature. Is there a doped-manganite compound, likely a large bandwidth one, that does not order at low temperatures and maintains a metallic character? Or, alternatively, can a metallic manganite compound become paramagnetic upon the application of, e.g., high pressure, at low temperatures? Searching for metal-insulator transitions near these quantum critical points, if they exist, may lead to surprises. If it is confirmed that, even in these circumstances, no superconductivity is found, then the S=1/2 spins of cuprates (as opposed to the higher spins of other materials) is likely to play a key role in the process. Studies by Riera and collaborators [11] have shown that the prominent presence of ferromagnetism and phase separation as the spin grows, may render superconductivity unstable in Mn oxides.
Technical applications of manganites remain a possibility. There are two areas of technologically-motivated investigations. One is based on materials with high T C , at or above room temperature, which may be useful as nearly half-metals in the construction of multilayers spin-valve-like devices. The group of Fert has made progress in this area recently, and there are several other groups working on the subject. Another area is the investigations of thin-films, exploiting CMR as an intrinsic property of these materials. Especial treatments of those films may still lead to a large MR at high temperatures and small fields, as needed for applications. Of course, there is a long way before CMR even matches the remarkable performance of the giant-MR (GMR) devices.

VIII. IS THE TENDENCY TO NANOCLUSTER FORMATION PRESENT IN OTHER MATERIALS?
We end this informal paper with a description of materials that present properties similar to those of the Mn-oxides, particularly regarding inhomogeneities, phase competition, and the occurrence of a T * (for a detailed long list of materials see [1]). It is interesting to speculate that all of these compounds share a similar phenomenology. Then, by investigating one particular system progress could be made in understanding the others as well. We start with the famous high-T c compounds, continue with diluted magnetic semiconductors -of much interest these days due to spintronic applications -and finish with organic and heavy-fermion materials.

A. Phase Separation in Cuprates
Considerable work has been recently devoted to the study of inhomogeneous states in cuprates. In this context, the issue of stripes as a form of inhomogeneity was raised several years ago after experimental work by Tranquada and collaborators and theoretical work by Kivelson, Zaanen, Poilblanc, and others. However, recent results obtained with STM techniques by Davis, Pan, and collaborators have revealed inhomogeneous states of a more complex nature. They appear in the surface of one of the most studied superconductors Bi2212 (but they may be representative of the bulk as well), at low temperatures in the superconducting phase, both in the underdoped and optimally doped regions. Figure 8 shows the spatial distribution of d-wave superconducting gaps, where the inhomogeneities are notorious. Indications of similar inhomogeneities have been observed using a variety of other techniques as well (the list of relevant experiments is simply too large to be reproduced here. For a partial list see [1]). Currently, this is a much debated area of research in high critical temperature superconductors, and it is unclear whether cuprates are intrinsically inhomogeneous in all its forms, or whether the inhomogeneities are a pathology of only a fraction of the Cu oxides. It is also unclear to what extend the notion of stripes, with at least partial order in its dynamical form, survives the Bi2212 evidence of inhomogeneities where the patches are totally random.  [12], using STM techniques. Shown on the left is a 600×600Å area. On the right a 150×150Å subset is enlarged. The gap scales are also shown.
Note that recent NMR work on YBCO by Bobroff et al. has not reported inhomogeneities, suggesting that phase separation may not be a generic feature of the cuprates. Similar conclusions were reached by Yeh and collaborators through the analysis of quasiparticle tunneling spectra. Clearly, more work is needed to clarify the role of inhomogeneities in the cuprate compounds.

B. Colossal Effects in Cuprates?
The results discussing the general aspects of the competition between two ordered phases in the presence of quenched disorder (see for instance the phase diagram in Sec.II) can in principle be applied to the cuprates as well, where superconductivity competes with an antiferromagnetic state (the latter perhaps including stripes). In fact, the phase diagram of the single-layer high-T c compound La 2−x Sr x CuO 4 contains a regime widely known as the "spin glass" region. In view of the results in [1,6], it is conceivable that this spin-glass region could have emerged, due to the influence of disorder, from a clean-limit first-order phase transition separating the doped antiferromagnetic state (again, probably with stripes) and the superconducting state. The resulting phase diagram would be as in Fig. 9(a). The grey region exhibits a possible coexistence of small locally-ordered clusters, globally forming a disordered phase. If this conjecture is correct, then small superconducting islands should exist in the spin-glass regime, an effect that may have already been observed by Iguchi et al. [13]. The results of Ong et al. on the Nernst effect may also find an explanation with this scenario.
If preformed superconducting regions are indeed present in underdoped cuprates, then the "alignment" of their order parameter (i.e. alignment of the local phases φ that appear in the complex-number order parameter ∆=|∆|e iφ for different islands) should be possible upon the influence of small external perturbations. This is similar to the presence of preformed ferromagnetic clusters in manganites, which align their moments when a relatively small external magnetic field is applied (see [6] and also Cheong and collaborators). Could it be that "colossal" effects occur in cuprates and other materials as well? This is an intriguing possibility raised by Burgy et al. [6]. There are already experiments by Decca and collaborators [14] that have reported an anomalously large "proximity effect" in underdoped YBa 2 Cu 3 O 6+δ . In fact, Decca et al. referred to the effect as "colossal proximity effect". Perhaps colossal-type effects are more frequent than previously expected, and they are prominent in manganites simply because one of the competing phases is a ferromagnet which can be favored by an easily available uniform magnetic field. If a simple FIG. 9: (a) Phase diagram of the 214 high temperature superconductor, as conjectured by Burgy et al. [6]. A possible first-order line (thick vertical line) separating the clean-limit superconducting and antiferromagnetic insulating regimes in the low hole-density region is shown. With disorder, the grey region is generated, exhibiting coexisting clusters. As in the manganite context discussion, T * is the remnant of the clean-limit ordering temperature. (b) Resistivity of the state conjectured in (a) using a crude resistor-network approximation, as calculated by Mayr. More details can be found in the original reference [6]. p is insulating fraction at 100 K. These fractions are also temperature dependent. external perturbation would favor charge-ordered, antiferromagnetic, or superconducting states, large effects may be observed in the region of competition with other phases. If this scenario is correct, several problems should be percolative, and studies by Mayr in [6] suggest that a resistor network mixing superconducting and insulating islands could roughly mimic results for cuprates (see Fig.9(b)). However, obviously more work is needed to test these very sketchy challenging ideas. Note that this scenario for cuprates is totally different from the "preformed Cooper pairs" ideas of Randeria and collaborators where the state above the critical temperature is homogeneous, but contains pairs that have not condensed. Here, the superconducting regions can be fairly large, containing probably hundreds of pairs, and are expected to move very slowly with time.
Even at a phenomenological level, the order of the transition between an antiferromagnet and a superconductor is still unknown. Mayr and collaborators are carrying simulations of toy models with both phases which hopefully will clarify these matters. The possibilities for the phase diagrams of two competing phases -according to general argumentations based on Landau free energies-involve bi-, tri-, and tetra-critical order. In the context of manganite models, the first two appear to be favored for the AF-FM competition since first-order transition at low temperatures have been found. However, in cuprates the tetracritical possibility in the clean limit has been proposed by Sachdev and collaborators as well. This would lead to local coexistence of antiferromagnetism and d-wave superconductivity in the underdoped region. Some experiments support this view [15]. Recent theoretical work also favors the emergence of tetracriticality from this competition [16], although a first-order transition is not excluded. More work is needed to clarify this elementary aspect of the problem.
The results for manganites and cuprates have also similarities with the physics of superconductivity in granular and highly disordered metals [17]. In this context, a critical temperature is obtained when the phases of the order parameters in different grains locks. This temperature is much smaller than the critical temperature of the homogeneous system. In manganites, the Curie or charge-ordering temperature may also be much higher in a clean system than in the real system, as discussed here and in [1,2].

C. Relaxor Ferroelectrics and T *
The so-called relaxor ferroelectrics are an interesting family of compounds, with ferroelectric properties and diffuse phase transitions [18]. The analogies with manganites are notorious. Representative cubic materials, such as PbMg 1/3 Nb 2/3 O 3 , exhibit a glass-like phase transition at a freezing temperature T g ∼230 K. Besides the usual features of a glassy transition, a broad frequency dependent peak in the dielectric constant has been found. The transition does not involve long-range ferroelectric order, providing a difference with most manganites that tend to have some form of long-range order at low temperature (although there are several with glassy behavior). However, a remarkable similarity is the appearance of another temperature scale, the socalled Burns temperature [19], which is ∼650 K in PbMg 1/3 Nb 2/3 O 3 . Below this temperature, polar nanoregions are formed. The Burns temperature appears to be the analog of the T * temperature of manganites discussed here. These analogies between ferroelectrics and Mn-oxides should be explored further. For example, is there some sort of "colossal" effect in the relaxor ferroelectrics? Are there two or more phases in competition? Can we alter the chemical compositions and obtain phase diagrams as rich as in manganites, including regimes with long-range order?

D. Diluted Magnetic Semiconductors and T *
Diluted magnetic semiconductors (DMS) based on III-V compounds are recently attracting considerable attention due to their combination of magnetic and semiconducting properties, that may lead to spintronic applications [20]. Ga 1−x Mn x As is the most studied of these compounds with a maximum Curie temperature T C ≈110K at low doping x, and with a carrier concentration p=0.1x [20]. The ferromagnetism is carrier-induced, with holes introduced by doping mediating the interaction between S=5/2 Mn 2+ -spins. Models similar to those proposed for manganites, with coupled spins and carriers, have been used for these compounds. Recently, the presence of a T * has also been proposed by Mayr et al. and Alvarez et al. [21], using Monte Carlo methods quite similar to those discussed in the manganite context. The idea is that in the random distribution of Mn spins, some of them spontaneously form clusters (i.e., they lie close to each other). These clusters can magnetically order by the standard Zener mechanism at some temperature, but different clusters may not correlate with each other until a much lower temperature is reached. This situation is illustrated in Fig.10. Naively, we expect that above the true T C , the clustered state will lead to an insulating behavior in the resistivity, as it happens in Mn-oxides (recent work by Alvarez has confirmed this hypothesis). This, together with the Zener character of the ferromagnetism, unveils unexpected similarities between DMS and Mn-oxides that should be studied in more detail.  [21]. At T * , uncorrelated ferromagnetic clusters are formed. At TC, they orient their moments in the same direction.

E. Cobaltites
Recent work has shown that cobaltites, such as La 1−x Sr x CoO 3 , also present tendencies toward magnetic phase separation [22]. NMR studies have established the coexistence of ferromagnetic regions, spin-glass regions, and hole-poor low-spin regions in these materials [23]. This occurs at all values x from 0.1 to 0.5. There are interesting similarities between these results and the NMR results of Papavassiliou and collaborators for the manganites, that showed clearly the tendencies toward mixed-phase states. The analysis of common features (and differences) between manganites and cobaltites should be pursued in the near future.

F. Organic Superconductors, Heavy Fermions, and T *
There are other families of materials that also present a competition between superconductivity and antiferromagnetism, as the cuprates do. One of them is the group of organic superconductors [26]. The large field of organic superconductivity will certainly not be reviewed here, but some recent references will be provided to guide the manganite/cuprate experts into this interesting area of research. In Fig.11 (left), the phase diagram [24] of a much studied material known as (TMTST) 2 PF 6 is shown. In a narrow region of pressures, a mixture of SDW (spin-density-wave) and SC (superconductivity) is observed at low temperatures. This region may result from coexisting domains of both phases, as in the FM-CO competition in manganites. The  FIG. 11: (left) Phase diagram of (TMTST)2PF6 [24]. SDW, M, and SC indicate spin-density-wave, metallic, and superconducting regimes. The gradient in shading indicates the amount of SC phase. (right) Phase diagram of κ-(ET)2X [25]. PNM means paramagnetic non-metallic, AFF a metallic phase with large AF fluctuations, and DWF is a density-wave with fluctuations. Note the first-order SC-AF transition (dashed lines), as well as the presence of a T * scale. Other details can be found in [25]. two competing phases have the same electronic density and the domains can be large. Alternatively, the coexistence of the two order parameters can be local, as in tetracritical phase diagrams. The layered organic superconductor κ-(ET) 2 Cu[N(CN) 2 ]Cl has also a phase diagram with coexisting SC and AF [27]. In addition, other materials of the same family present a f irst-order SC-AF transition at low temperatures, according to the phase diagram recently reported in [25]. This result is reproduced in Fig.11 (right), where a characteristic scale T * produced by charge fluctuations is also shown. The similarities with results for manganites are strong: first-order transitions or phase coexistence, and the presence of a T * . It remains to be investigated whether the similarities are accidental or reveal the same phenomenology observed in competing phases of manganites.
FIG. 12: Schematic temperature-pressure phase diagram of some Ce-based heavy fermions, from [28]. The notation is standard.
Another family of materials with competing SC and AF are the heavy fermions. As in previous cases, here just a few references are provided to illustrate similarities with other compounds. In Fig.12, a schematic phase diagram of Ce-based heavy fermions is shown [28]. Note the presence of a pseudogap (PG) region, with non Fermi-liquid (NFL) behavior at higher temperatures. The PG phase is reminiscent of the region between ordering temperatures and T * , discussed in manganites and cuprates, and it may arise from phase competition. In addition, clusters in U-and Ce-alloys were discussed in [29], and spin-glass behavior in CeNi 1−x Cu x was reported in [30]. Recently, Takimoto et al. [31] found interesting formal analogies between models for manganites and for f -electron systems that deserve further studies.

IX. CONCLUSIONS
In recent years a large effort has been devoted to the study of manganese oxides. Considerable progress has been achieved. Among the most important unveiled aspects is the presence of tendencies toward inhomogeneous states, both in experiments and in simulations of models. However, considerably more work is needed to fully understand these challenging compounds and confirm the relevance of nanoclusters for the CMR effect.
Considering a more global view of the problem, recent years have brought a deeper understanding of the many analogies among a large fraction of the materials belonging to the correlated-electron family. The analogies between organic, cuprate, and heavy-fermion superconductors are strong. The SC-AF phase competition with the presence of a T * , pseudogaps, phase coexistence, and first-order transitions, is formally similar to analogous phenomena unveiled in the FM-CO competition for manganites. In addition, all of these compounds share a similar phenomenology above the critical temperature, with nanoclusters formed. This phenomenon occurs even in diluted magnetic semiconductors, relaxor ferroelectrics, and other families of compounds. The self-organization of clustered structures in the ground state appears to be a characteristic of many interesting materials, and work in this promising area of investigations is just starting. This "complexity" seems a common feature of correlated electron systems, particularly in the regime of colossal effects where small changes in external parameters leads to a drastic rearrangement of the ground state.

X. ACKNOWLEDGMENTS
The author thanks S. L. Cooper, J. A. Fernandez-Baca, G. Alvarez, and A. Moreo, for comments and help in the preparation of this manuscript. The work is supported by the Division of Materials Research of the National Science Foundation (grant DMR-0122523).