Adventures in Crystal Growth: Synthesis and Characterization of

Review pubs.acs.org/cm

Adventures in Crystal Growth: Synthesis and Characterization of Single Crystals of Complex Intermetallic Compounds W. Adam Phelan, Melissa C. Menard, Michael J. Kangas, Gregory T. McCandless, Brenton L. Drake, and Julia Y. Chan* Department of Chemistry, Louisiana State University, 232 Choppin Hall, Baton Rouge, Louisiana 70803, United States S Supporting Information *

ABSTRACT: The central motivation of this manuscript is to highlight the discovery of novel, highly correlated electron systems. Ternary phases with unusual ground states composed of lanthanides, transition metals, and main group elements exhibit competing behavior leading to potentially novel physical properties. These systems are known to exhibit exotic properties such as unconventional superconductivity, heavy fermion behavior, and unusual forms of magnetism. The flux-growth method provides an avenue to discover and grow large single crystalline materials so that structure−physical property relationships may be determined. The growth of high quality single crystals is necessary to elucidate the intrinsic magnetic, electronic, and thermodynamic properties and has played a significant role in advancing basic and applied materials research. In this manuscript, we review the crystal structures, physical properties, and structure−property correlations of a select group of intermetallic compounds to demonstrate the potential for growth and discovery of materials using main group flux. KEYWORDS: intermetallics, crystal growth, gallides, alumnides, antimonides, highly correlated systems, magnetism,Yb3Pd2Ga8

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INTRODUCTION Intermetallic compounds exhibiting strong electron−electron or electron−lattice interactions can give rise to fascinating and unusual phenomena. These strongly correlated electron materials exhibit interesting physical properties such as unusual forms of magnetism and unconventional superconductivity that are mediated by thermal and quantum fluctuations. The growth of high quality single crystals is necessary to study the physics of correlated materials,1 tune the properties of existing materials by growing related analogues, and to discover new systems which will lead to the understanding of the chemistry and physics of materials that will impact technology, as discussed in a recent National Academy of Science report entitled, “Frontiers in Crystalline Matter: From Discovery to Technology”.2 Having the desire to fundamentally understand the physics and chemistry of correlated materials and to unlock their complete potential, our group and others have been involved in the exploratory synthesis, characterization, and elucidation of crystal structure-physical property relationships of novel intermetallic compounds for the past decade. Special attention has been paid to multinary systems composed of lanthanide-transition metal-Groups 13−15 elements. The coupling of the conduction electrons to the localized or itinerant behavior of the 4f or nd electrons of the rare-earth and transition-metal elements can result in systems which display long-range magnetic order, heavy-fermion behavior, and unconventional superconductivity. Traditionally, solid state growth techniques for growth of intermetalllics involve high reaction temperatures (up to 2273 K). During the past 10−15 years, low temperature routes such as © 2011 American Chemical Society

hydrothermal, microwave, and molten salts methods have been employed to discover many new compounds. To elucidate intrinsic materials properties with respect to crystallographic directions, the growth of sufficiently large single crystal growths of intermetallics is preferred. Coupled with the need for discovery of materials in areas of alternative energy, defense, information technology, and the availability of new experimental methods to measure physical properties, the growth of single crystals is a key to sustainability. The self-flux-growth technique3,4 has been employed to synthesize many phases exhibiting interesting physical properties. The flux-growth procedure can be advantageous compared to traditional solid state techniques. Benefits of this method include kinetic and thermodynamic control of product formation. Perhaps, the most exciting advantages are the potential growth of defect-free, phase pure, large single crystals. However, successfully growing large single crystals of a desired phase can present many challenges including limited temperature ranges, competing phases, and homogeneity ranges. Solutions to a number of challenges are described along with the compounds studied, where appropriate. A selected collection of crystals synthesized using the flux-growth method is shown in Figure 1. This review will highlight physical properties, the experimental variables for the preparation of single crystal phases using low melting Group 13−15 fluxes, the challenges and future directions for Received: July 12, 2011 Revised: November 21, 2011 Published: December 15, 2011 409

dx.doi.org/10.1021/cm2019873 | Chem. Mater. 2012, 24, 409−420

Chemistry of Materials

Review

Figure 1. Selected crystals synthesized using the flux-growth method.

Figure 2. Schematic of a reaction ampule and temperature profile detailing the growth of CePdGa6, Ce2PdGa10, and Ce2PdGa12, which serve to highlight the subtleties of parameter choices for crystal growth. considered, and published binary phase diagrams should be consulted to choose reaction profile to avoid the formation of undesirable stable phases. The removal of excess flux from the single crystal surface can be accomplished by mechanical separation, sonicating the crystals in hot water, or chemical etching mixtures, such as DMF/I2 with dilute HCl or HNO3 or NaOH. An example of reaction profile highlighting the synthesis of various gallide phases is shown in Figure 2. Single crystals are usually characterized by a combination of powder and single crystal X-ray diffraction for phase identification, phase purity, and structure determination. Elemental analysis by energydispersive X-ray spectroscopy (EDXS) is performed on single crystals to ensure homogeneity and to determine their chemical composition. Upon elucidation of crystal structures, physical property measurements (magnetic susceptibility, field dependent magnetization, electrical resistivity, and heat capacity measurements) are conducted on the same batch of single crystals. If possible, it is preferred to have single crystal diffraction, elemental analysis, and physical properties performed on the same single crystal to fully understand structure− property relationships.

the growth of materials, and last will provide some final thoughts and conclusions.

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EXPERIMENTAL SECTION

This section is intended to give a general description of the growth of single crystals by the flux-growth method and general characterization of the compounds described in each section. More details regarding the growth of each compound can be found in cited references within the respective sections of this manuscript. The flux-growth technique overcomes reaction surface area by using molten elements. The melting point depression that results from the combination of flux with the other reactant metals allows for the search for new materials at intermediate temperatures as low as 500−700 K and eliminates the necessity of annealing samples for long periods (∼ several weeks) to obtain congruently and incongruently melting products. The details of the flux-growth method have been previously reported3,4 and will briefly be reviewed. Metals, in the form of powders or small ingots, are typically weighed out in the desired reaction ratio and loaded into a crucible. This crucible is sealed in an evacuated fused-silica tube and backfilled with an inert gas (if necessary). After sealing, the fused-silica tube is set inside a larger crucible (to hold the reaction vessel upright) and then placed into a furnace. Adjustable variables that can affect phase formation and/or crystal size include the heating/cooling rates, the dwell times at high/low temperatures, the dwell temperatures, and the sample removal temperature. At the completion of the reaction profile, samples are either spun in a centrifuge to remove excess flux, quenched in water, or set on the benchtop to cool. There are several issues to consider when choosing a flux for the growth of intermetallics. The flux should have a relatively low melting point and a high boiling point, such as Ga, Sn, Sb, and Al. The reactivity of the flux with the other reactants should be carefully

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GALLIDES AND ALUMINIDES The ternary intermetallic compounds crystallizing in the HoCoGa5 structure type5 and related structure types have garnered tremendous attention. The CeMIn5 (M = Co, Rh, Ir) family of compounds are categorized as an interesting class of materials where the superconducting mechanism is not necessarily mediated by electron−phonon interactions, but in fact is magnetically mediated.6−8 CeCoIn5, CeIrIn5, and CeRhIn5 superconduct at low temperatures, and their heavy fermion states are characterized by their enhanced Sommerfeld 410



Review

coefficients γ of 290, 720, 420 mJ mol-Ce−1 K−2, respectively. The Sommerfeld coefficient γ is the electronic contribution of heat capacity where Cp = γT + βT3. Heavy fermion behavior is often associated with the valence instability of the 4f and 5f electrons of Ce, Yb, Eu, or U containing compounds and has a large Sommerfeld coefficient (γ) typically two orders greater than a typical metal like copper (∼1 mJ mol−1 K−2).9−12 Ce2RhIn8 (γ ∼ 400 mJ mol-Ce−1 K−2) and Ce2IrIn8 (γ ∼ 700 mJ mol-Ce−1 K−2) possess similar Sommerfeld coefficients to CeRhIn5 and CeIrIn5, and a superconducting state exists only for Ce2RhIn8 with application of magnetic field and pressure.13 Ce−Pd−Ga Phases. Although relatively few Ce−Pd−Ga compounds have been reported, several compounds are of particular interest. CePd2Ga3 is a ferromagnetic Kondo lattice system, and despite the characteristic screening of magnetic moments by conduction electrons, the compound orders ferromagnetically at 6 K.14 For more details on Kondo physics, please refer to the review articles by Doniach, Löhneysen et al., and Yang et al.15−17 More recently, the growth and physical properties of three new Ce−Pd−Ga ternary have been reported. Figure 2 shows the reaction profile for obtaining CePdGa6,18 Ce2PdGa10,19 and Ce2PdGa12.20 The reaction ratios for the synthesis of all three phases are roughly 1:1−1.5:20 with respect to Ce:Pd:Ga, and product formation is dependent on the cooling rates and spin temperature. Although it would be useful to determine an appropriate heat treatment and optimize conditions for the gallium growths with differential thermal analysis (DTA), gallium is very reactive with the Pt cups typically used as sample holders in DTA experiments. Al2O3based cups can also be used for thermal analysis. However, DTA experiments are inhibited for lanthanide-rich containing materials because of the reactivity of reactants with the Al2O3 crucible. A method involving DTA measurements for volatile metallic samples using sealed Ta crucible is described by Janssen et al.21,22 and can certainly be advantageous for designing temperature profiles for syntheses of compounds similar to intermetallics described in this review. The crystal structures of CePdGa 6, Ce2PdGa10, and Ce2PdGa12 are shown in Figure 3, and the subunits present

the number of Ce−Ga contacts and Ce−Ce interatomic distance can be drawn for CePdGa6 and Ce2PdGa10. Fewer Ce−Ga contacts in Ce2PdGa10 compared to CePdGa6 are consistent with reduced hybridization between magnetic moments and conduction electrons, and Ga-only tetrahedral slabs in Ce2PdGa10 effectively separate Ce moments along the c-axis. These observations regarding Ce2PdGa10 are consistent with the reduced ordering temperature when compared to CePdGa6. Ce2PdGa12, isostructural to Sm2NiGa12,27 can be described as a stacking of CePdGa6 units separated by Ga-only segments.24 Ce2PdGa12 exhibits a double magnetic transition, with Ce3+ moments ordering antiferromagnetically at 11 K along the caxis and ferromagnetic ordering at 3 K in the ab-plane because of the canting of spins. Again, a correlation has been suggested for Ce2PdGa12 and CePdGa6 regarding the number of conduction electrons and Ce−Ga hybridization. When compared to CePdGa6, the higher ordering temperature (TN ∼ 11 K for Ce2PdGa12) and smaller Sommerfeld coefficient (γ ∼ 140 mJ mol-Ce−1 K−2) in Ce2PdGa12 is consistent with more conduction electrons leading to higher ordering temperature. This also illustrates the proposed mechanism that local electronic environments, which can also influence hybridization effects between 4f and conduction electrons leading to stronger RKKY interactions (Ruderman−Kittel−Kasuya−Yosida).9,24,28 Coupled with the potentially rich properties that are found in the Ce−Pd−Ga phases, similar structures can be compared to illuminate the effects of crystal chemistry on electrical and magnetic properties. Ln2MGa12 (Ln = La−Nd, Sm; M = Ni, Cu). Several isostructural analogues of Ln2MGa12 (Ln = La−Nd; M = Ni, Cu, Pd) were grown to investigate the effects of structural stability and carrier density by transition metal substitution in the Ln2MGa12 phases.29−31 The structure Ce2PdGa12 is shown in Figure 3, and compounds adopting the Ce2NiGa12 structure type crystallize in the tetragonal P4/nbm space group with lattice parameters of ∼6 × 6 × 15 Å3. Powder X-ray diffraction patterns indicate that La 2 CuGa 12 is isomorphous to Sm2NiGa12,27 and subsequent powder neutron measurements led to modeling of the static disorder. Following the growth of the Ln2CuGa12 (Ln = Pr, Nd, Sm), the occupancy of Cu decreases with decreasing Ln ionic radii. The end member of this series appears to be the Sm analogue, and the latter lanthanide analogues are not stabilized in this structure type. In addition, we note that the occupancy of the transition metal decreases with the smaller lanthanide analogues. With the exception of Ce2CuGa12, the Ln2MGa12 (Ln = La−Nd; M = Ni, Cu, Pd) phases order antiferromagnetically and are strongly field-dependent, consistent with the large magnetoresistance (MR % = ρH − ρo)/ρo × 100%) observed in many of the Ln2MGa12 phases.29,31 The Ni-analogues order at higher temperatures than the Cu-analogues, which could be attributed to stronger RKKY interactions between the 4f and conduction electrons in Ni-analogues and to stronger screening of the 4f electrons by the conduction electrons in the Cu-analogues, where Kondo behavior predominates. Ln4MGa12 (Ln = Tb−Er, M = Fe, Pd, and Pt). With the latter lanthanides, compounds adopt the Y4PdGa12 structure type. Single crystals of Ln4MGa12 (Ln = Y, Tb−Er, M = Pd, Pt) were grown using the reaction ratio 1:1:20 while determining the structural stability of Ln−Pd−Ga phases. The samples were heated at 1423 K for 7 h, and then cooled to 803 at 15 K h−1.29,32 Similar conditions were reported for the synthesis of

Figure 3. Crystal structures of CePdGa6, Ce2PdGa10, and Ce2PdGa12.

in related gallides have been reviewed by Grin.23 Specific heat and magnetic measurements indicated CePdGa6 is a heavy fermion compound with γ ∼ 300 mJ mol−1 K−2, and the onset of antiferromagnetic ordering below 5.5 K is indicated by cusps in the magnetic susceptibility when the field is applied along both the ab-plane (2.48 μB) and c-axis (2.45 μB).24 Paramagnetic Ce2PdGa10, isostructural to Ce2NiGa10,25 exhibits a large change in the electrical resistivity (∼ 200%) as a function of an applied magnetic field up to 9 T.26 A correlation between 411



Review

the isostructural Y4Mn1−xGa12−yGey.33 However, in the attempt to grow suitable crystals of Ln4FeGa12 (Ln = Y, Tb−Er) for physical properties, a polycrystalline sample must first be prepared and then annealed with a seed crystal and excess gallium flux, as both direct attempts at flux-growth and arcmelting yielded mixed phases.34 The Ln4MGa12 compounds crystallize in space group Im3̅m with lattice parameters a ∼ 8.5 Å adopting the Y4PdGa12 structure type,35 a variant of the U4Re7Si6 structure.36 The crystal structure, shown in Figure 4,

phenomena, such as heavy fermion behavior, superconductivity, and magnetism.9,43,44 It is important to note both SEM/EDXS and ICP-OES indicate that Cu and/or Ga are deficient in this structure, which was predicted for BaCuxAl13−x.45 The deficiency optimizes Cu−Ga contact distances, lowering the overall energy of the structure. After careful consideration of the literature and crystal structure, it was determined that the Cu position (8b) was partially occupied, resulting in a final refined composition of ∼ Ln(Cu,Ga)12.8 from X-ray diffraction data, and the structure is shown in Figure 5.

Figure 4. Crystal structure of Y4PdGa12.

Figure 5. Crystal structure of Ce(Cu,Ga)13.

can be described as a large cube formed by stacking eight LnGa3 (AuCu3 structure type) unit cells. In LnGa3, the rare earth atoms occupy the corners of a cube, and the gallium atoms occupy the faces. For the ternary phase, the transition metals occupy 1/4 of the octahedral holes formed by the gallium atoms forming a body centered cell. Incorporation of a transition metal slightly distorts the Ga position of the LnGa3 binary subunit.37 In Er4Fe0.67Ga12, the partial occupancy of the Fe site allows for the modeling of gallium in both its ideal and its disordered position.34 An additional type of disorder was reported in Tb4Ag0.9Ga12.1, where mixing of gallium onto the transition metal site was observed.38 In addition, modulation is observed in Y4 Mn 0.74 Ga 12 , but not in the quaternary Y4Mn1−xGa12−yGey phase.39 Ln4MGa12 (Ln = Tb−Er, M = Pd, Pt) compounds exhibit metallic resistivity. All analogues are observed to order antiferromagnetically between 3 and 16 K, except for Ho4PdGa12, which shows no magnetic ordering down to 2 K. In addition, Tb4PdGa12, Tb4PtGa12, Er4PdGa12, and Er4PtGa12 show meta-magnetic transitions at 5, 0.3, 1, and 1 T, respectively. Ln4MGa12 (Ln = Dy−Er, M = Pd, Pt) compounds show fairly large positive magnetoresistance, with the palladium compounds having a change in resistivity between 10 and 40% at 3 K and 9 T. Under the same temperature and magnetic field conditions, the platinum compounds show much larger magnetoresistance of 50%, 220%, and 900% for Dy, Er, and Ho, respectively.40 Heat capacity measurements have been reported for a number of uranium analogues with the transition metals Fe, Co, Rh, and Pd and show enhanced electron mass behavior with the Sommerfeld coefficients ranging between 80 to 140 mJ mol−1 K−2.37,41 Ln(Cu,Ga)13−x (Ln = La−Nd, and Eu). Understanding the rich chemistry found for Ln−Pd/Pt−Ga phases, it was of interest to determine if isostructural analogues could be stabilized and how a systematic substitution of Cu for Pd or Pt would impact the physical properties. During exploration of the Ln−Cu−Ga phase space, the Ln(Cu,Ga)13−x (Ln = La−Nd, and Eu) phases crystallizing in the well-known NaZn13 structure type were grown.42 Interest in this structure type arises from a combination of highly correlated electron

Ln(Cu,Ga)13−x (Ln = Ce, Pr, Nd, and Eu) shows no evidence for long-range magnetic order down to 2 K; however, Weiss temperatures indicate antiferromagnetic correlations for the Ce, Pr, and Nd analogues. As evidenced by the effective moment and unit cell volume, the Eu is in the divalent state with a positive Weiss temperature indicative of ferromagnetic coupling. A broad shoulder is observed in the temperature dependence of the resistivity for Pr(Cu,Ga)13−x because of Kondo coherence, a low temperature state where the localized f-electrons become itinerant and hybridize with the conduction electrons forming a narrow conduction band that is characterized by a heavy charge carrier mass.46 Interestingly, Pr(Cu,Ga)13−x shows large positive magnetoresistance, which leads to quasiparticle mass enhancement due to strong electron correlations47,48 with Pr(Cu,Ga)13−x as one of the few examples of a nonsuperconducting heavy electron system. SmCu4Ga8. Sm containing compounds are of interest because of their potential for valence instabilities and fluctuations. Unlike the other lanthanide analogues, SmCu4Ga8 adopts the SmZn11 structure type crystallizing in P6/mmm space group with lattice parameters of a ∼ 8.9 Å and c ∼ 8.6 Å.49 SmCu4Ga8 is structurally related to the CaCu5 structure type, and the structure is shown in Figure 6. SmCu4Ga8 orders

Figure 6. Crystal structure of SmCu4Ga8.

antiferromagnetically at 3.3 K, and the linear inverse magnetic susceptibility (down to 10 K) indicates that the van Vleck contribution due to crystal field splitting is either very small or nonexistenta rather uncommon feature for Sm containing compound coupled with minimized crystal electric field effects. 412



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This is illustrated with β-LnNi1−xGa4 (Ln = Tb−Er) which also exhibits structural disorder in the Ni/Ga network. In this case, the occupation of Ni at the tetrahedral (4̅m2) site results in occupational disorder in the immediate surroundings (BaAl4type LnNi1−xGa3 segment).67 The degree of structural order in a phase is linked to the stability of the phase as illustrated by both Ce2Ag1−xGa10−y and β-LnNi1−xGa4 (Ln = Tb−Er). The growth of these disordered phases is accomplished in a very narrow temperature range, with profiles similar to that of CePdGa 6 with only minor differences in the cooling sequence.18 Ce2Ag1−xGa10−y was cooled initially at a rate of 200 K h−1 to 773 K, then slow cooled at 2 K h−1 to 673 while βLnNi1−xGa4 (Ln = Tb−Er) was also cooled to 773 K then later cooled to 723 K at a rate of 8 K h−1. Slower cooling between 773 and 673 K is necessary to avoid the growth of more stable, structurally ordered phases such as BaAl4-type Ce(Ag,Ga)468 and a reduced amount of α-LnNiGa4. To discover new phases with structural disorder, we can utilize both the coordination preference as an adjustable parameter and a two-step cooling sequence (fast and then slow prior to dwelling at low temperature) to avoid structurally stable phases. Ln(Cu,Al)12 (Ln = Y, Ce−Nd, Sm, Gd−Ho, and Yb). With large single crystals readily attainable for Ln(Cu,Ga)13−x,42 SmCu4Ga8,29 and Ln(Cu,Ga)12,53 it was of interest to see if isostructural analogues could be grown with the use of an Al flux. This should provide insight into the role a main group element plays on the structural stability and the physical properties of the materials. Though the aluminum analogues of the NaZn13 and SmCu4Al8 could not be stabilized, Ln(Cu,Al)12 of the ThMn12 structure type (Figure 7) was stabilized for Ln = Y, Ce−Nd, Sm, Gd−Ho, and Yb with stoichiometry close to LnCu4.6Al7.4.53 Although Ln(Cu,Al)12 (Ln = Ce and Pr) does not order magnetically down to 3 K, the heat capacity data for the Ce analogue reveals a transition that was attributed to an antiferromagnetic order consistent with the negative Weiss (θw ) temperatures observed from the molar magnetic susceptibility. Sm(Cu,Al)12 orders antiferromagnetically at 5 K and with no evidence of van Vleck contribution to the susceptibility, a feature also absent in SmCu4Ga8. Low temperature heat capacity data reveals that Ln(Cu,Al)12 (Ln = Ce, Pr, and Sm) show enhanced mass behavior with γ ∼ 390, 80, and 120 mJ mol-Ln−1 K−2, respectively. In contrast to Ln(Cu,Ga)12, the aluminum analogues do not show large magnetoresistance (less than 10% effect), indicating either Ga or the larger degree of structural disorder in Ln(Cu,Ga)12 plays an important role in the large positive magnetoresistance observed in Ln(Cu,Ga)12. Ln(Ag,Al,Si)2 (Ln = Ce and Gd). The search for systems with specific qualities can often lead to unexpected discoveries. With the complex disorder observed in Ln2Ag1−xGa10−y, it was of interest to see if this disorder would be observed in isostructural aluminum analogues. During the exploration of the Ln−Ag−Al system, single crystals of Ln(Ag,Al,Si)2 (Ln = Ce and Gd) were prepared serendipitously.73 SEM/EDXS revealed that Ln, Si, Al, and Ag were present in the phase, as the Si was extracted from the silica wool filtering medium. High quality single crystals were grown with Ln:Ag:Al:Si in a molar ratio of 1:1:10:1.2. LnM2 (Ln = Ce and Gd and M = Ag, Al, and Si) crystallizes in the tetragonal α-ThSi2 (I41/amd, a ∼ 4.2 Å and b ∼ 14.4 Å) structure type with the 4a and 8e Wyckoff positions occupied by Ln and M atoms, respectively. The structure, shown in

Ln(Cu,Ga)12 (Ln = Y, Gd−Er, and Yb). The Ln(M,Al)12 (M = Cr, Mn, Fe or Cu) compounds elicit interesting physical properties50 ranging from enhanced mass behavior in CeCu4Al8 and CeCr4Al8,51−53 and both lanthanide and transition metal magnetic ordering in ErFe4Al8,54−57 mixed valency in Yb(Cu,Al)12,53 and negative magnetoresistance in LnFe4Al8 (Ln = Sc, Y, Ce, Yb, and Lu).58,59 Ln(Cu,Ga)12 crystallizes in the tetragonal ThMn12 structure type (I4/mmm, a ∼ 8.75 Å and c ∼ 5.13 Å) with only the Ln and Cu having dedicated Wyckoff positions of 2a and 8f, respectively. The remaining two positions in the unit cell, 8i and 8j, are statistically disordered with Cu and Ga resulting in a composition close to LnCu5.5Ga6.5 and the structure of the compound is shown in Figure 7.

Figure 7. Crystal structure of Ce(Cu,Al)12.

The Ln(Cu,Ga)12 phases order antiferromagnetically (12.5, 13.5, 6.7, and 3.4 K for Gd−Dy, and Er, respectively) with the exception of Ho, which remains paramagnetic down to 3 K. The ordering temperature scales with the de Gennes factor, dG = (gj − 1)2J(J + 1), a scaling factor that can be used to compare the magnetic ordering temperatures of lanthanide containing materials based on angular momentum.28 Large positive magnetoresistance (MR), up to a maximum of 117%, 127%, 150%, 105%, and 141% with an applied field of 9 T at 3 K, is found for the Y, Gd, Dy−Er analogues, respectively. Most notably, the nonmagnetic Y(Cu,Ga)12 shows large positive MR similar to the magnetic rare earth analogues. Ce2Ag1−xGa10−y and β-LnNi1−xGa4 (Ln = Tb−Er). The search for structurally disordered phases is driven by the link between disorder and unusual magnetic and electronic behavior attributable to the structural modulation of LnCoxGa3Ge (Ln = Y, Gd),60,61 to the spin density wave formation in U3Ga2Si3 because of disorder in the Ga/Si-network,62,63 and the negative thermal expansion of YbGa1+xGe1−x coincident with the puckered Ga/Ge nets.64 An introduction to density waves can be found in the review article written by Brown and colleagues.65 The investigation of defect-variants of the Ce2NiGa10 structure type, Ce2Ag1−xGa10−y,66 and β-LnNi1−xGa4 (Ln = Tb−Er)67 illustrate the effects of coordination preference and atomic size on structural order. Since Ag is usually found in a tetrahedral environment for intermetallics,66,68−70 structural disorder is observed in the Ag/Ga network in Ce2Ag1−xGa10−y. The physical effects resulting from the disordered Ag/Ganetwork in Ce2Ag1−xGa10−y were observed in the magnetic susceptibility measurements resulting in the spin glass-like magnetic behavior. Similar behavior has been observed in Ce2CuSi3 and Ce2CuGe3, where the mixing of a transition metal (Cu) with a main group element (Si or Ge) resulted in the spin glass-like behavior.71,72 There are two Ni environments in β-LnNi1−xGa4 (Ln = Tb−Er): a structurally ordered rectangular prism (Ni CN = 8) and a distorted tetrahedron. 413



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Figure 8, can be described as an open three-dimensional (3-d) channel network of M atoms with the Ln atom occupying the

and low temperature magnetic order and heavy fermion behavior of Ce3Co4Sn13.97−99 Both compounds were grown using excess tin flux which seems ideal given tin’s low melting temperature. However, given the thermodynamic stability of the LnSn3 binaries and the propensity to form Sn inclusions, the use of tin as a flux can be problematic. The first problem is easily overcome by removing the sample above the melting temperature of the LnSn3. Etching crystals with dilute HCl can often remove the tin on the crystal surfaces and leave the product intact.3 Tin inclusions in samples are often seen in the low temperature resistivity, as tin superconducts below 3.7 K. Fortunately, the upper critical field of tin is small in magnitude and this signal can be suppressed by the application of a weak magnetic field. The following section describes the structure elucidation and the physical properties Ce3Co4Sn13. Ce3Co4Sn13. Ce3Co4Sn13 adopts the Yb3Rh4Sn13 structure type100 crystallizing in the cubic space group Pm3̅n with lattice parameters of a ∼ 9.6 Å, V ∼ 900 Å3. The crystal structure of Ce3Co4Sn13 is depicted in Figure 9 and related to the double

Figure 8. Crystal structure of CeM2 (M = Ag, Al, and Si).

interstitial sites found in these channels. The Ln occupies the center of a 12-coordinate polyhedron with 8 M atoms forming a rectangular prism about the Ln atom. The rectangular prisms are capped above and below by two additional M atoms forming triangular prisms situated orthogonal to the other. The polyhedra pack in a space filling, face sharing arrangement. M−M and Ln−M contacts agree well with related Ce(Al,Si)2, Yb(Ag,Si)2, and CeSi2 phases.74−76 Compounds of the α-ThSi2 structure type are of interest because of the Kondo behavior observed in the parent phase CeSi2−x.77,78 Additionally, many of the parent phases (CeSi2−x, CeGe2, and LnAlxSi2−x) order magnetically and strongly dependent on dopant concentration.79−85 With complex magnetic, Kondo, and heavy fermion behavior in related phases, it was of interest to determine how Ag disorder throughout the lattice would impact the physical properties. Ferromagnetic order is less common for Ce-containing intermetallics, but has been observed among the subset of Ce compounds adopting the α−ThSi2 structure type.78,80,83,85,86 CeM2 (M = Ag, Al, and Si) orders ferromagnetically at ∼11 K in both H∥c and H∥ab with a subsequent antiferromagnetic modulation at 8.8 K observed in the molar magnetic susceptibility for H∥ab. Heat capacity measurements reveal two maxima at low temperature which are in good agreement with the ferromagnetic transition and subsequent antiferromagnetic modulation observed in the molar magnetic susceptibility. It is important to note that the antiferromagnetic modulation observed here has previously been described in CeSi1.70.78,87,88 GdM2 orders antiferromagnetically at 24 K with H∥c with no ordering observed down to 3 K for H∥ab. A fit of the data to the modified Curie−Weiss law reveals large negative Weiss temperatures for both the H∥ab and the H∥c directions. The large negative θW for H∥ab results in frustration values >25, significantly larger than the range of 2−5 (θW/TN) for typical antiferromagnetic materials.89 Additionally, resistivity as a function of temperature can best be described by the Kondo model where a minimum in the resistivity is observed for both CeM2 and GdM2. The impact of ∼0.1 mol of Ag substitution is rather dramatic, suppressing the antiferromagnetic transition by 8 K over that observed for Gd(Al,Si)2.83 Stannnides. The chemistry and physics of stannides have been extensively studied because the bonding networks range from covalent to pseudoionic depending on the stannide compound and were recently reviewed.90,91 With respect to the physics of stannides, the LnRhxSny compounds have been widely studied, particularly with respect to superconductivity and magnetism.92−96 The reinvestigation of LnRhxSny compounds led to the discovery of superconductivity in La3Co4Sn13

Figure 9. Crystal structure of Ce3Co4Sn13.

perovskite, A′A″3B4X12, where the Sn atoms occupy the A′ and X sites, the Ln atoms occupy the A″ sites, and the Co atoms occupy the B sites. Ce3Co4Sn13 is built up of Sn1(Sn2)12 icosahedra, face- and edge-sharing Ce(Sn2)12 cuboctahedra, and corner-sharing Co(Sn2)6 trigonal prisms. More descriptively, the infinite network of edge-sharing Sn1(Sn2)12 icosahedra pack in a CsCl arrangement, and the faces of each icosahedron are connected to 8 Co(Sn2)6 trigonal prisms and 12 Ce(Sn2)12 cuboctahedra.99 The Ce cuboctahedral environment is common in several magnetically mediated superconducting heavy fermion intermetallics including CeMIn5 (M = Co, Rh, Ir)101−103 and Ce2MIn8104 (M = Rh and Ir). Since the Ce coordination environment of Ce3Co4Sn13 is also found in CeMIn5 (M = Co, Rh, Ir) and Ce2MIn8 (M = Rh and Ir), single crystals of Ce3Co4Sn13 were prepared to determine and compare their physical properties. Perhaps the most notable feature of Ce3Co4Sn13 is the rather large Sommerfeld coefficient, γ ∼ 4200 mJ mol-Ce−1 K−2, derived from heat capacity measurements which is indicative of heavy fermion (enhanced electron mass) behavior at low temperatures.99 A similar value was confirmed by heat capacity measurements for polycrystalline samples of Ce3Co4Sn13.105 A peak at ∼0.65 K in the heat capacity was attributed to antiferromagnetic order consistent with the negative Weiss temperature obtained from Curie− Weiss fits to the magnetic susceptibility data for single crystals of Ce3Co4Sn13.99 However, neutron diffraction experiments between 0.4 K−2.0 K showed no signs of long-range order.106 More recent neutron scattering experiments confirmed the emergence of short-range antiferromagnetic order induced by the application of a magnetic field. In these experiments, the 414



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Table 1. Antimonides Growth Conditions structure type

space group

CeCrSb3 CeNiSb3 (α) CeNiSb3 (β)

Pbcm Pbcm Pbcm Pbcm Pbcm P4/nmm Cmca

HfCuSi2 SmSb2

stability Ln Ln Ln Ln Ln Ln Ln

= = = = = = =

La−Nd, Sm, Gd−Dy; M = V, Cr La−Nd, Sm; M = Ni La−Ce; M = Ni, Pd Pr, Nd, Sm, Gd, Tb; M = Ni Pr, Nd, Sm, Gd, Tb; M = Fe Y, Gd−Er; M = Ni, Cu La−Nd, Sm

dwell T (K)

spin T (K)

flux

1453 1373−1423 1423 1423 1373 1423 1448

1123 943 523 573 913 943 1023

Sb Sb Sn Sb with Sn substitution Sb Sb Sb

common orthorhombic space group Pbcm and have similar a and b lattice dimensions (a ∼ 12 −13 Å and b ∼ 6 Å) and is distinguishable by the c-dimensions. These three polymorphs, as shown in Figures 10 a−c, can be described as being built up

(001) peak, a symmetry forbidden hkl reflection, was found to increase in intensity with the application of a magnetic field.107 Antimonides. Zintl phases containing antimony are known to exhibit a wide variety of interesting, exotic, and diverse bonding networks including planes, ribbons, and chains.108−110 Antimony containing phases are also known to also exhibit diverse physical properties such as high thermoelectric figures of merit,111−114 colossal magnetoresistance,115 and heavyfermion behavior.116 Recent reports of Zintl compounds for thermoelectric applications111,117,118 and the rich structural chemistry have also been reviewed.119 Reduced crystal dimensionalities often lead to exotic physics as magnetic and quantum fluctuations are usually enhanced. This combined with rich two-dimensional (2-d) structural chemistry mentioned above serves as motivation to prepare and characterize new or unexplored antimonide phases. The inherent volatility of antimony complicates synthesis using traditional arc-melting techniques. A large number of thermodynamically stable binary-line compounds often are stabilized making lower temperature metastable phases more difficult to prepare.3 Additionally, removal of excess antimony from yielded crystals requires aqua regia which can be damaging to the product crystals if not careful. The use of a tin flux and the stoichiometric combination of the desired elements can circumvent the formation of binary antimonide phases. However, similar X-ray scattering form factors of tin and antimony can pose challenges for structure determination using laboratory X-ray techniques. Table 1 summarizes the structural stability and synthetic conditions for synthesis of selected layered lanthanide antimonides. This section outlines the synthetic strategies, structural characterization, and physical properties. LnSb2. The light diantimonides LnSb2 (Ln = La−Nd, Sm) crystallize in the orthorhombic SmSb2 structure type with lattice parameters of a ∼ 6 Å, b ∼ 6 Å, and c ∼ 18 Å and is built up of alternating layers of La/Sb and Sb sheets.120,121 Synthetic details used to obtain large single crystals of LnSb2 have been described elsewhere.122 Early electrical resistivity measurements confirmed that LaSb2 behaves as metal down to low temperatures, and the onset of superconductivity was observed at 0.4 K.123 Magnetic susceptibility and transport measurements later showed that the diamagnetic LaSb2 is highly anisotropic with large residual resistivity ratio (RRR) values despite of its micaceous nature indicative of the quality of crystals. Compounds of this structure type show anomalously large magnetoresistance122 (up to 100-fold linear increase in resistance between 0 and 45 T for the La analogue).121 More recently, the dimensionality of the superconducting state was found to change from 2-d to 3-d with the application of pressures greater than 2 kbar.124 LnMSb3 (M = V, Cr, Fe, Ni). The ternary LnMSb3 (M = V, Cr, Fe, Ni) compounds described herein crystallize in a

Figure 10. Crystal structure of (a) CeCrSb3, (b) β-CeNiSb3, (c) αCeNiSb3, and (d) YNi1−xSb2.

of Ln layers separated by a nearly square net of Sb atoms, ∞[Sb], which are sandwiched between layers of M-centered octahedra, 2∞[MSb2].125−127 The main structural distinction for the polymorphs of LnMSb3 lies in the packing of the 2∞[MSb2] octahedral layer along the c-axis. LnMSb3 (Ln = La−Nd, Sm, Gd−Dy for M = V and Ln = La−Sm, Gd, Yb for M = Cr) adopts the LaCrSb3 structure type with a ∼ 13, b ∼ 6, and c ∼ 6 Å while c-axis dimension of the structurally related phases βLnNiX3 (Ln = La−Nd, Sm, Gd, and Tb; X = Sn, Sb) and LnFeSb3 (Ln = Pr, Nd, Sm, Gd, and Tb) is doubled to ∼12 Å and the c-axis of α-LnNiSb3 (Ln = Ce, Pr, Nd, and Sm) is tripled with c ∼ 18 Å. The 2∞[MSb2] octahedra in compounds adopting the LnCrSb3 structure type are edge-sharing in the [010] direction and face-sharing in the [001].132 The 2∞[MSb2] octahedra in βLnNiSb3 and LnFeSb3 are edge-sharing in [010], whereas they are face-sharing, with every other octahedron sharing edges in [001] giving rise to an approximate doubling of the c-lattice parameter, ∼12 Å. The 2∞[NiSb2] octahedra in α-LnNiSb3 are edge-sharing in the [010] direction and both edge- and facesharing in the [001] direction, resulting in an approximate tripling of its c-lattice parameter, ∼18 Å.126,127 The inability for latter lanthanide analogues of α-LnNiSb3, β-LnNiX3, and LnFeSb3 to be stabilized in the structure types as described has been attributed to the increasing distortion within the 2

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Chemistry of Materials 2

Review

prepared using arc-melting techniques.142−144 LnNi1−xSb2 (Ln = Y, Gd−Er) compounds crystallize with the HfCuSi2 structure type in the tetragonal space group P4/nmm with lattice parameters of a ∼ 4 Å and c ∼ 9 Å,145 and the structure of YNi1−xSb2 is highlighted in Figure 10d. The layered LnNi1−xSb2 structure closely resembles the layered structures of LnSb2, β-LnNiSb3, and β-LnNiX3. In this structure, the Ln atoms are inserted between a layer of Sb atoms forming a square net and a NiSb4 tetrahedral layer. The Ln atoms are coordinated to 8 Sb atoms adopting a distorted square antiprismatic geometry (4 Sb atoms from the Sb net and 4 Sb atoms in the NiSb4 tetrahedral layer). Several late transition metals, such as Ni, Cu, Pd, and Ag, and Cd, can also be stabilized in this structure type.146 Polycrystalline LnNi1−xSb2 (Ln = Gd−Ho) order antiferromagnetically with Née l temperatures higher than the corresponding single crystals of LnNi1−xSb2 (Ln = Gd−Ho), thus emphasizing the importance of synthesizing high quality, large single crystals to determine their intrinsic behavior.142−144 The experimental effective moments obtained from Curie− Weiss fitting of susceptibility data reveal that the Ln sublattice contributes solely to the magnetism in compounds in this structure type. The temperature-dependent electrical resistivity confirmed that all analogues are metallic down to 2 K and large positive magnetoresistance (up to 100% at 9 T) was observed for LnNi1−xSb2 (Ln = Y, Dy, Ho).

and 2∞[MSb2] layers because of the lanthanide contraction. Magnetic anisotropy is an important, reoccurring theme in the magnetic behavior of the LnMSb3 (M = V, Cr, Ni, Fe).131,133−135 The LnCrSb3 (Ln = magnetic lanthanide) compounds125 consist of two magnetic sub-lattices where the magnetic moments of Cr order ferromagnetically with ordering temperatures between TC ∼ 105−125 K. At low temperatures, the lanthanide moments order antiferromagnetically.133,136,137 Studies on single crystals of LaCrSb3 and SmCrSb3 show that magnetic easy axis was found to lie within the bc-plane just below the Curie temperatures.133,135 The most intriguing analogue is the YbCrSb3, which orders ferromagnetically at Tc ∼ 280 K and is exclusively due to the Cr moments.138 Single crystals LnVSb3 (Ln = Ce−Nd, Sm, Gd−Dy) are also magnetically anisotropic with the Ce analogue ordering ferromagnetically while all other lanthanide analogues order antiferromagnetically at low temperatures with vanadium not bearing a magnetic moment.139 Magnetic susceptibility measurements show that α-CeNiSb3 orders ferromagnetically with a TC ∼ 6 K, while α-PrNiSb3, α-NdNiSb3, and α-SmNiSb3 order antiferromagnetically with TN = 4.5, 4.6, and 2.9 K, respectively.126,127 Anisotropic behavior for the latter three analogues was observed with the magnetic easy axis corresponding to the a-direction.127 α-CeNiSb3 is metallic and shows Kondo lattice behavior near 25 K. The ferromagnetic transition temperature increases with applied pressures up to 25 Kbar, then decreases for P > 25 kbar, and the onset of a second ordering temperature emerged between 35 kbar ≤ P ≤ 55 kbar. A quantum critical point at P ∼ 60 kbar was suggested because of the nearly complete suppression of the first magnetic phase and an increase in the T2 coefficient and ρo evidenced from resistivity at this pressure.140 It is worth noting that α-CeNiSb3 and β-CeNiSb3 order ferromagnetically around 6 K, but antiferromagnetic order is not observed in the remaining analogues of β-LnNiSb3 (Ln = Pr, Nd, Sm, Gd, and Tb) unlike the antiferromagnetic order observed in α-LnNiSb3 (Ln = Pr, Nd, and Sm) analogues. The temperature dependent electrical resistivity data show that the β-LnNiSb3 (Ln = La and Ce) and β-LnNiX3 (Ln = Pr, Nd, Sm, and Gd; X = Sb and Sn) analogues also exhibit metallic behavior and exhibit positive magnetoresistance, with the magnetoresistance of β-CeNiSb3 saturating above 50% at 8 T and 3 K.130 Large positive MR in β-LnNiSb3 (Ln = La and Ce) and β-LnNiX3 (Ln = Pr, Nd, Sm, and Gd) strongly resembles the LnSb2 (Ln = La, Pr, Sm, and Nd).122 Heat capacity and resistivity of β-CeNiSb3 confirm the emergence of Kondo lattice behavior at low temperatures.130 Magnetic susceptibility of LnFeSb3 except TbFeSb3 (Ln = Pr, Nd, Sm, and Gd) is highly anisotropic and scale with the de Gennes factor. The temperature-dependent electrical resistivity measured along the bc-plane for LnFeSb3 (Ln = Pr, Nd, Sm, Gd, and Tb) indicated that these compounds are metallic. Kinks in the resistivity of PrFeSb3, NdFeSb3, and TbFeSb3 coincide with the antiferromagnetic order observed in these compounds. PrFeSb3 exhibits positive nonsaturating magnetoresistance up to 100% at 9 T and T = 3 K with the remaining analogues exhibiting small positive magnetoresistance under these conditions.131 LnNi1−xSb2 (Ln = Y, Gd−Er; x ∼ 0.4). The desire to prepare Ln−Ni-Sb ternary compounds with heavy lanthanides yielded LnNi1−xSb2 (Ln = Y, Gd−Er; x ∼ 0.4). Polycrystalline samples of (Ln = La−Nd and Sm−Ho) have previously been ∞[Sb]

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CURRENT CHALLENGES The physical and chemical properties of an intermetallic can be vastly altered or tuned by the substitution of the lanthanide. On the basis of the success in growing and characterizing a number of Ln−Pd−Ga phases (Ln = La, Ce, Tb−Er) described earlier, the Yb−Pd−Ga system provides a new frontier to explore. In addition, ytterbium-containing intermetallics are particularly attractive because they can exhibit unusual properties including valence instability and superconducting heavy fermion behavior.147,148 As part of the effort to grow Yb-containing phases, Yb, Pd, and Ga elements with purities greater than 99.9% were weighed out in the molar ratio 1.5:1:15, heated at a rate of 150 K h−1 to 1423 K, and dwelled for 7 h. The samples were then rapidly cooled (150 K h−1) to 773 K and, finally slow cooled to 673 K at a rate of 8 K h−1. The samples were then inverted and centrifuged to separate the crystals from excess gallium flux. The resulting crystals were metallic silver in color and typical dimensions were ∼2 × 2 × 3 mm3, as shown in Figure 11. The composition from EDXS measurements was found to be Yb2Pd2.5(3)Ga8.1(3) and was normalized to the lanthanide concentration. Several structural models have been evaluated to solve the crystal structure of Yb2Pd3Ga9 with crystallographic parameters, and the details for the four most suitable models are provided in the Supporting Information, Table S1. Difficulties in the refinement led to literature search for other structurally related 2−3−9 intermetallic compounds. This search yielded a list of compounds adopting the orthorhombic Y2Co3Ga9 structure type149 (Cmcm, No. 63) which have been reported for the ternary aluminides Ce2Rh3Al9,150 Gd2Ir3Al9,151 Nd2Co3Al9,152 and U2Co3Al9,153 and the ternary gallides Y2Rh3Ga9,154 Eu2Rh3Ga9,155 Eu2Ir3Ga9,155 Dy2Ru3Ga9,156 Ho2Co3Ga9,149 and Lu2Rh3Ga9.157 The models for these compounds share the same number of unique and fully occupied Wyckoff atomic sites with one rare earth site (8g), two transition metal sites (4a and 8e), and four Group 13 element sites (4c, 8f, 8g, and 16h). 416



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CONCLUSIONS AND FUTURE WORK During the past several years, new highly correlated Ce and Ybbased compounds have been discovered. These intermetallics have been grown as large single crystals using gallium, indium, antimony, tin, and aluminum fluxes, and many of them show valence instabilitiescharacteristic of heavy fermion behavior. Understanding puzzling correlations such as the interplay of magnetism and superconductivity, which is quantum in nature, is only possible with high quality single crystal growth. Efforts are now being focused on Yb based materials because Yb2+ or Yb3+ ions can hybridize with the conduction electrons leading to highly correlated states. This hybridization has been beautifully illustrated in mixed valent β-YbAlB4159 where quantum criticality develops without the application of chemical or physical stimuli.148 In the continued interest to explore systems near the quantum critical regions of their magnetic phase diagrams, our group and others seek to discover compounds where the electron−electron interactions are strong and, therefore, collective quantum mechanical effects are not only observable but also can control the basic material properties that have significant technological implications. Although quantum criticality has traditionally been studied as a low temperature phenomenon,160 the behavior could persist even at ambient temperatures.161 Understanding the interplay of quantum phase transitions may someday provide a new mechanism for exotic ground states exhibiting interesting electrical and magnetic properties.162 To achieve the goals for studying the chemistry and physics of new materials, high quality single crystals must be grown to fully characterize the system.

Figure 11. Aggregate crystals of Yb3Pd2Ga9.

Unfortunately, the ordered Y2Co3Ga9 model could not be adopted as a good fit for the data collected of single crystal of Yb2Pd3Ga9 with a crystal mosaicity of 0.439(3)°. On the basis of the suggested systematic absences, space group Cmcm (Laue class mmm) has the highest figure of merit with cell parameters a ∼ 13.2 Å, b ∼ 7.6 Å, c ∼ 9.5 Å, V ∼ 951 Å3. Although a model based on the ordered Y2Co3Ga9 structure type did not provide an acceptable fit, the data collected from the Yb2Pd3Ga9 single crystal could be modeled with positional disorder in a similar fashion as described for Ho2Rh3Al9 and Er2Ir3Al9 by Niermann et al.158 Unlike the published structures for Ho2Rh3Al9 and Er2Ir3Al9, the final refined occupancies of the atomic sites involved in the translational position disordered layers were left as free variables in our models without any constraints, and the atomic positions are provided in the Supporting Information, Table S2. The least-squares refinement converged with a R1 of 0.030, Rint of 0.093, Δρmax of 2.53 e Å−3, and Δρmin of −2.98 e Å−3. In addition, the goodness of fit parameter, S, is 0.89 after applying a weighting scheme of w = 1/[σ2(Fo2) + (0.0419P)2] where P = (Fo2 + 2Fc2)/3. There are two partially occupied Ga sites within the misplaced Yb2Ga3 layers that have been identified as “non-positive definite” in the list of principal mean square atomic displacements U which is probably caused by both the low partial site occupation (∼32%) and the large uncertainties for the anisotropic displacement parameters (i.e., the uncertainty is the same size, or close to the same size, as the calculated displacement). Of the four models provided in the Supporting Information, the orthorhombic Cmcm model with positional disorder appears to be the best fit of the data. Niermann et al. stated for their Ho2Rh3Al9 and Er2Ir3Al9 compounds “that both structure determinations and refinements were not straightforward”.158 This sentiment rings true in the case of solving the structure of Yb2Pd3Ga9, and the disordered orthorhombic model is very likely the most suitable choice. This example shows some of the complexities that arise in elucidating structures of these types of phases, and detailed microscopy methods will certainly confirm the structures. Further low temperature physical property measurements are in progress to elucidate the physical properties of this Yb compound.

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ASSOCIATED CONTENT S Supporting Information * Additional commentary on the evaluation of the four most reasonable structural models considered for Yb2Pd3Ga9 as well as 10 separate tables detailing the crystal structure information obtained via X-ray diffraction. Additionally, four separated .cif files have been submitted for this compound. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Phone: 225-578-2695. Fax: 225-5783458.

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ACKNOWLEDGMENTS The authors acknowledge F. R. Fronczek and physics collaborators for useful discussions and Alfred P. Sloan Fellowship, NSF Grants DMR 0756281 and 1063735 for partial support of this project.

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