1. Detailed Project Description: Optical spectroscopy of Rare Earth Compounds, Environmentally friendly pigments

Figure 1.  Rare earth based pigments are already used in commercial applications. We propose to understanding their optical properties starting from first principles. The goal is to design materials with gaps spanning the whole visible spectra.

Motivation  Lanthanide based compounds are the subject of current intensive investigations due to their fundamental scientific interest and their numerous practical applications. Cerium oxide is now attracting significant attention because of its catalytic properties as the CeO CeO reaction is used in the catalytic converters of vehicles. The optical properties of rare earth oxides (as well as compounds closely related to rare earths oxides and sulfides  [21,22,23,24,25,26,27,29,28,30]) are used to color objects ranging from cars to children toys, replacing the more toxic Cd based pigments as illustrated in fig. 1.

These materials combine the physical properties of a Mott insulator and those of a band insulator. The -electrons are localized while the rare earth -states are empty and the O p states are occupied. A schematic picture of the density of states is sketched in figure 2. The Ce  band is split by correlations into a lower and upper Hubbard band while the oxygen  states broaden into a valence band while the Ce  states broaden into a conduction band hence this material embodies an interesting combination of the physical properties of Mott insulator and band insulators.

Previous Work  Numerous electronic structure methods have been applied to this problem. For example Refs.[31,32] use LDA treating the -electron as core, Ref.[33] applied a hybrid of B3LYP calculation of antiferromagnetic CeO and finally the work of our French collaborators Windiks et. al. [34] applying the LDA+U method, and Pourovskii et. al. [35] who applied LDA+DMFT to CeO.

Proposed Research None of these approaches is entirely successful. Density functional theory based methods fail to describe the Mott insulating state above the Neel temperature which in CeO is very low. LDA+DMFT describes well the local moment regime above the Mott insulator but fails account quantitatively for the magnitude of the  gap which is severely underestimated in LDA.

The challenge to electronic structure theory is to predict both the rare earth  gap and the  gap starting from first principles, and then to use these insights to explain the chemical trends and design materials with given chromatic properties. This requires the quantitative evaluation of the relevant optical conductivity matrix elements.

As a first step we propose to neglect the current vertex corrections (they are not vanishing even in the limit of infinite coordination in multiorbital models but are expected to be small when exciton effects are not important), the expression for the optical conductivity becomes

                                                 (1)

where we have denoted , and used the shortcut notations , . The matrix elements  resemble the standard dipole allowed transition probabilities which are now defined with the right and left solutions  and  of the Dyson equation:

                                   (2)

where we have denoted  ,  and assumed that  while. . Here  are velocities of electrons and are the Kohn-Sham like solutions of the Dyson equation.

The current matrix elements can be evaluated directly in a given basis set, and this implementation within LDA+DMFT has been carried by K. Haule and V. Oudovenko at Rutgers [36,37] (for a review see Kotliar et. al.  [9]). An alternative to this, is provided by the Peierls substitution which replaces the matrix elements of the gradient by  with  the Kohn-Sham Hamiltonian. The advantage of this approach is its simplicity and the fact that it can be used in an arbitrary basis set, once the one electron part of the Hamiltonian is available. This approach was implemented by G. Palsson [38] using the LMTART [39] as a basic code. Recently J.M. Tomczak, [40] has developed this approach in connection to the downfolding procedure used in the Stuttgart LMTO code [41]. The quality of the Peierls approximation is basis set dependent and its accuracy depends on the material. While this fact is well known, (see for example [42,43,40]), there are very few system specific studies of its accuracy [44], and preliminary investigations indicate that in CeO there can be substantial discrepancies between these approaches.

We propose to compare the direct evaluation of gradients and the evaluation of the current matrix elements via the Peierls substitution within the same basis set to asses the quality of the latter for the rare earth oxide materials. In addition, we will derive formulae to improve the accuracy of the Peierls substitution. Evaluations of these corrections would not only allow us to determine the reliability of the Peierls substitution determination of the current matrix elements, but could be useful for investigating in which basis this substitution is most accurate. The technical developments in this section will also be useful for the thermal transport studies described in the next section [45]. Since the issues of the determination of the matrix elements of the thermal current via generalizations of the Peierls approach also arise in the context of thermal transport [42].

Figure 2.  Schematic one particle density of states of the rare earth oxides. They combine the character of band and Mott insulators since correlations split the -level into a lower and upper Hubbard band. The -level moves deeper as one progress into the rare earth series.

Figure 3.  left) Measured dielectric function of two rare earth sulfide compounds [27]. CeS shows a small peak preceding (probably connected to  to  transitions) the a large rise connected to the oxygen  to Ce- transitions. right) Preliminary DMFT results for the optical conductivity of CeO displaying qualitatively the same type of spectra.

Dynamical mean field theory captures the physics of the Mott transition and properly describes the multiplet structure of the -electron which in the rare earth oxides survives in the solid, as we have shown in collaboration with the Ecole Polytechnique group (LDA+DMFT work in progress with S. Biermman L. Pourovskii and A. Georges).

The LDA+DMFT approach however does not resolve the difficulty of the underestimation of the gap between the  states of the oxygen and the unoccupied -states of the rare earth. It is well known that the LDA exchange and correlation potential severely underestimates this quantity and LDA+DMFT inherits this deficiency. To overcome this problem we propose to build on the success of hybrid functionals. By combining the B3LYP exchange and correlation potential which has been shown to give excellent results for gaps in a number of compounds ( see for example, [46]) with dynamical mean field theory we expect to obtain an overall accurate electronic structure of these compounds. This approach is more economic than the full implementation of GW+DMFT which is also being pursued by both groups with different collaborators [15,16].

We will focus first, for simplicity on the LaO compound and the CeO compounds that allows us to distinguish between materials where the  electron plays a crucial role and one where the -level is empty. After validating the accuracy of the proposed approach against experimental data on these simple systems we will proceed with the study of other oxides  and sulfides [22,25,23,26]. We will also investigate the less explored family of fluorosulfides  [28,29,30] with the chemical formula SF, where  is a rare earth material. Of particular interest is SmSF where multiplet splitting might control the onset of optical absorbtion.

Theory will deliver the chemical trends of the oxides across the rare earth series. We will proceed with investigating related compounds which have not yet been synthesized in the lab, such as CeNdO. Hence we will be to suggest materials that have optical gaps spanning the whole optical spectrum filling an important void in this area.

2. Strongly Correlated Thermoelectrics

1. Background on Thermoelectrics

Electrons in solids carry both charge and entropy and hence can conduct both heat and electricity. Thermal and electrical currents are coupled together, and thermal gradients can be used to induced voltages. This thermoelectric coupling can be used to construct devices that act as refrigerators, power generators or temperature sensors. Thermoelectric devices are attractive for many applications as they have no moving parts, use no liquid refrigerant and last for a long time. The major disadvantage of current thermoelectric devices is their poor efficiency.

The efficiency of a material for thermoelectric applications, is found to depend on material properties through the dimensionless parameter , called figure of merit:

                                                                        (3)

where  is the absolute temperature,  is the electrical conductivity,  is the Seebeck coefficient or thermopower, and  is the total thermal conductivity. The total thermal conductivity can be decomposed into two parts,  where  is the heat carried by the electrons and holes and  is the heat carried by the phonons. The thermoelectric power factor is defined by  is a good indicator of the thermoelectric potential of a material when the thermal conductivity is dominated by the lattice contribution. Since the lattice thermal conductivity cannot be smaller than its electronic component, a good thermoelectric requires a Seebeck coefficient larger than 300 .

All three solid state properties (,  and ) are determined by the details of the electronic structure of the material. For weakly correlated semiconductors, these are the band gap, band shape, and band degeneracy near the Fermi level and scattering of charge carriers (electrons or holes) and thus are not independent variables. The same is true for correlated materials, except that the description of the electronic structure requires many body notions, since sometimes the currents are not carried by quasiparticle bands, well described by band theory, but by carriers in Hubbard bands which are better described in real space than momentum space.

One important application of thermoelectric is to power generation. At the heart of the generators is the thermoelectric material, which converts thermal energy of electrons and holes into electric current. Other thermoelectrics applications such as the cooling of detectors, or freon free refrigerators are required to have an optimal figure of merit at room temperature, or even at very low temperatures.

Most of present day commercial thermoelectric materials were discovered in 1950s with figure of merit () between 0.4 and 1.0. They are based on semiconductors, in particular n-type BiTe, BiSe and p-type SbTe. However, significant progress in improving  was achieved in last two decades by identifying new materials with enhanced thermoelectric properties.

Hicks and Dresselhaus [47,48] have shown that quantum-well superlattice structures result in an improvement in the figure of merit of various semiconducting compounds. Also artificial superlattice thin-film structures grown from chemical vapor deposition, such as BiTe/SbTe [49], and by molecular beam epitaxy, such as PbSeTe/PbTe [50,51,52], have been introduced with substantially enhanced  values relative to those of their bulk counterparts. Marking an important development in this area, specially constructed BiTe/SbTe superlattices were reported to exhibit a very high  of 2.4 at room temperature [53]. Substantial theoretical work has been done recently based on model type calculations showing a great potential of improving thermal response by constructing superlattices [54,55,56,57,58,59,60,61,62,63,64].

Another road for the improvement of the figure of merit, involves the study of some strongly correlated materials. Several surprising and unexpectedly large improvements in the figure of merit have been obtained in materials such as chalcogenides [65], half-Heusler alloys [66], skutterudites [67,68], metal oxides [3,69,70], intermetallic clathrates [71,72,73], and pentatellurides [74]. For instance, considerable interest was generated recently by the class of strongly correlated materials called filled skutterudites with general chemical formula  where  is a lanthanide atom (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Yb),  is Fe, Ru, Os and  is P, As or Sb. The open crystal environment of skutterudite compounds typified by the presence of structural voids offers an exciting opportunity to modify the material by inserting foreign species into the voids and thus dramatically alter phonon transport reducing the phonon contribution to thermal conductivity and increasing the figure of merit [75]. With the judicious choice of the filler species one can prepare both n- and p- type filled skutterudites that display very high thermoelectric figure of merit in excess of 1.4 in CeFeCoSb around 1000K [76].

Figure 4.  left.a)  Crystal structure of FeSb showing Fe (brown crosses) surrounded by Sb (blue octahedra). The two short Fe-Sb bond distances are shown in black and white. left.b) The Fe-Sb-Fe bond angle associated with the edge sharing octahedra along the c-axis of the unit cell. right) Real part of the optical conductivity as a function of temperature of FeSb in the infrared range, with light polarized perpendicular to the b crystallographic axis. The arrow indicates the onset of the temperature independent tail, ascribed to the incoherent electronic contribution. Insets: reflectivity and conductivity at 10 and 150 K for frequencies up to 105 . Reproduced from Ref. [78]

Figure 5.  Thermoelectric power factor  as a function of temperature () measured in magnetic fields () of 0T (open data points with dotted curves) and 9T (filled data points with solid curves), applied along the a-axis (black squares), b-axis (red circles) and c-axis (blue and green triangles). Reproduced from Ref. [77]

More recently, a surprisingly large Seebeck coefficient was found by Bentien et. al. [77] in the marcasite FeSb, whose structure is shown in figure 2. The experimental results for the power factor shown in figure 3 are two orders of magnitude higher than in any previously known compounds. These experimental results on a remarkably simple binary compound were completely unexpected, and the mechanism for this large thermoelectric response is not understood. The phenomena occurs at relatively low temperatures, and is probably related to other anomalies such as the development of a gap in the optical conductivity at those temperatures, as shown in figure 2-right. The strong dependence of the effect on magnetic fields suggests that strong correlations are also present in this compound.

The thermoelectric figure of merit in this compound is limited by its high thermal conductivity as discuss by Bentien et. al. Therefore it is very likely that using this compound in multilayers, that can reduce the thermal conductivity, will result in further dramatic improvements. However this basic research proposal will focus first on elucidating the physical mechanism for the high thermoelectric response in this material. We will approach the material design problem of identifying the mechanisms for the high thermoelectric response and to answer the question of what are the scales that set the temperature scale for the maximum of the power factor in this surprisingly simple compound.

However we notice that  is not just a curiosity but can serve as the “fruit fly" of correlated thermoelectrics. Understanding the origin of the large power factor in this relatively simple material will teach us important lessons about the mechanisms behind the large thermoelectric response in strongly correlated systems, before facing the challenges more complex materials containing cages and rattlers to reduce the thermal conductivity of the compound.

The tools to be developed in this proposal however are quite general and will allow a more systematic exploration theoretically more systematically, the prospects of good thermoelectric responses in materials with more complex unit cell. In the last year of funding we will tackle the problem of thermoelectricity in the misfit cobaltates [79,80] and in the rare earth filled skutterudites [75,76].

1. Theoretical Background on Correlated Thermoelectricity

Strongly correlated compounds require treatments beyond scope of traditional electronic structure methods [81,82,83,84,85,86,87,88]. Some time ago our group at Rutgers also addressed the problem of thermoelectricity [89,90] applying DMFT to simplified model and showed that the figure of merit can be substantially increased as a result of correlations for example near a Mott transition. These studies used model Hamiltonian studies and illustrated the potential of strongly correlated materials and to demonstrate the power of the theoretical approach, the Dynamical Mean Field Theory.

The transport coefficients that govern the thermopower, electric and thermal conductivity can be expressed in terms of the matrix of kinetic coefficients  relating the electric and thermal currents ,  to the applied external fields , . Transport quantities become , , . The thermoelectric response thus reduces to the evaluation of kinetic coefficients.

In strongly correlated materials, transport has the coherent and the incoherent contribution. If current vertex corrections can be ignored, the expression for the kinetic coefficients becomes

                       (1)

where  are velocities of electrons and  is the electron spectral density of the multiorbital spectral function

                                                     (2)

In the limit of weak correlations the quasiparticle picture becomes valid in terms of a relaxation time. Noticing that in this limit , is a Lorentzian function, the above equation reduces to the more familiar expressions of the kinetic coefficients

                                                         (3)

Proposed Research

Until recently the research on thermoelectricity of correlated materials were based on model Hamiltonians to evaluate the spectral function, and ignored vertex corrections which are potentially important in the multiorbital case. Vertex corrections are potentially important for the thermoelectric response than for the optical conductivity since vertex corrections are likely to be less relevant at high frequencies. Simplified model Hamiltonians tend to ignore important information which is very relevant to the thermoelectric response such as the degree of particle hole asymmetry in a given band structures. We propose to remove these two shortcomings in the calculations of the Seebeck coefficient and the dc conductivity.

We will obtain system specific information by using the LDA+DMFT approach, the Green’s function can be expressed by

                                                          (4)

where  are the Kohn-Sham like solutions of the Dyson equation.

Within LDA+DMFT the local vertex function can be computed from the solution of the impurity. This in turn can be used to compute the full correlation function of the problem [7].

To compute the thermoelectric properties of FeSb, several theoretical developments have to take place. First a DMFT electronic structure calculation of the FeSb has to be carried out. We will first do an all electron DMFT calculation with the correlations on the Fe atom. This involves a multiband calculations, for which the recently developed Continuous time QMC [91] will be used. Then we will evaluate the optical conductivity with the tools exemplified above (extension of Eq. (1) to finite frequencies). The fact that the optical conductivity qualitatively resembles that found in earlier model studies of the Anderson lattice model suggest, that this might be an adequate approach to the physics of the material. Other alternatives simpler models might emerge by downfolding to few band model.

A second step will be the evaluation of the static and frequency dependent thermal transport. For that we will implement recent advances in the derivation of the form of the thermal current of interacting systems [42,92]. To evaluate the thermal transport we will require improvements in the analytic continuation techniques at low frequency. We will explore the possibility of combining maximum entropy method at high frequency and polynomial approximations at low frequency. With this tools in hand we will perform an evaluation of the Seebeck coefficient as a function of temperature. We will also contrast it with the high frequency behavior of this coefficient. This quantity is easier to calculate, but might not reflect the true low frequency transport quantity.

Finally after validating our procedure on the FeSb system, we will address the challenge of material design by exploring what are the important factors that determine the scale of the gap. We will do that by varying in the computation the strength of the Sb-Fe hybridization, and the number of -electrons which we suspect to be the key quantities.

Guided by these insight we will then address the problem of optimization of the figure of merit by substitutions of various transition metal ions that will accomplish changes in the valence and hybridization. The material design goal that we seek to achieve is to stabilize the hybridization gap at higher temperatures and to increase the power factor. We will study the effects of substibuting Co for Fe and As and Bi for Sn while keeping ordered solid state structures , for example, CoSb CoSbBi and CoAs.

Finally with the experience gained from these studies and the new theoretical tools which we will have develop in this project we will investigate compounds with complex unit cells which naturally reduce the thermal conductivity. We will consider the misfit cobaltites [79,80] and the skutterudites discussed in section ?.

1. Educational Component

In parallel with the research efforts the lab without walls will serve a meaningful educational purpose, that will reach a broad community. .

We will create a virtual laboratory for correlated materials and a database of theoretical studies of materials using Dynamical Mean Field Theory. These tools will be accessible via the internet, and will be part of a new form of international collaboration which will realize the vision of a "lab without walls". Computer infrastructure and internet will play several important roles. First, it will be the media containing the virtual laboratory. Second, after this work is completed it will allow public access to the new databases and theoretical spectroscopy tools.

These tools will help educate not only experts in many body theory but also to non experts, material scientists, experimental physicists and chemists, who would like to compare their experimental results with DMFT calculations.

This lab will also include a separate set of pedagogical tools, to teach basic concepts to undergraduate students, for example how to think about metals and insulators in terms of the density of states (a many body concept). Their purpose is to help non-experts understand properties of correlated materials. For example, traditionally band structure diagrams are used in material science to represent electron’s energy and momenta graphically. However, the spectral function representation might be more helpful because it can be used for strongly and weakly correlated materials and is more closely related to experimental probes such as ARPES. Animations and simulation using the materials which we will be studying will be developed by graduate students and postdocs funded by this grant. Euginia Etkina from the Graduate School of Education has expressed interest in evaluating these tools and pilot testing them in her course "Multiple Representation in Physical Science" which is a course taught every year in the School of Education.

Viktor Oudovenko, the information technology specialist in the physics department in collaboration with the student and postdocs associated to this proposal will help the PI’s direct the education and information technology efforts of building the lab without walls.

To help disseminate the DMFT tools developed in this project, this four year program will culminate with a hands-on training workshop focused on applying the new generation of DMFT tools to practical problems. The location of the concluding workshop has not yet been decided but we anticipate to submit proposals to both the ICTP in Trieste and to the Boulder Summer School to host this event. We mention that a workshop of this kind organized by V. Anisimov and G. Kotliar was among the first, and that numerous schools of this kind have been corganized since by our French partners and the European -network. This will be allow American graduate students and post-graduate researchers to participate in this highly effective training program.

2. Management Plan and Schedule of Visits

The collaboration has been quite productive on an informal basis with frequent visits of the PI’s to each others institutions as interesting physics develops. In addition, we have installed telecommunication facilities using Skype and cameras and the internet which allows day to day communication between the parties. We also use the most recently developed programs allowing to share desktops of computers (for better visualization and demonstration) as well as world resources which allow fast file sharing (if different groups in the world need access to the same files). In addition to this infrastructure we have planned one visit a year where the complete team meets. The meeting will alternate one year in the US and one year in Paris.

The lab without walls will be launched in July 2008 with the funding of this proposal together with the corresponding French counterpart to the CNRS. The first meeting of the team will take place when the French comes to the US to participate in the Aspen workshop on realistic treatments of strongly correlated electrons. A. Georges and O. Parcollet are coorganizers of that event. The second year meeting will take place in France, July 2009 to evaluate the qualitative physics of the novel thermoelectrics to be studied in this proposal and the evaluation of the thermoelectric response. The meeting will be arrange to coincide with the French GDR meeting on thermoelectrics headed by Charles Simon. The third year meeting will take place at Rutgers in July 2010, when the strategy for optimizing thermoelectricity will be finalized in the light of the results obtained the previous year. The fourth meeting will take place March 2011 to summarize the achievement as well as to propose further steps for the continuation of the collaboration and the lab without walls .

Most of the funding requested is to support one postdoctoral associate who will be directly associated to all the phases of this project. The postdoctoral associate will be supervised jointly by the French and the US team, and is expected to spend a substantial fraction of his time abroad. To cover his travel expenses we have budgeted foreign travel. In addition, French resources from the Pascal Chair that was awarded to G. Kotliar and from the CNRS will be used to support those extended visits.

We summarize the goals of each year of the proposal and the corresponding deliverables below.

· First Year: Development of the computational tools to evaluate the optical response. Evaluation of the optical conductivity of CeO. LDA+DMFT study of the one electron spectra of SbFe.

· Second Year: development of the computational methods needed to evaluate the thermoelectric response. Evaluation of the optical conductivity of the rare earth oxide series as well as the rare earth fluorides. Exploration of compounds with gaps that cover the electromagnetic spectrum.

· The third year will be devoted to the exploration of the improvement of the thermoelectric figure of merit, starting with computations of the Seebeck coefficient of SbFe and related compounds.

· The fourth year will apply the insights and the tools developed in the previous years to compounds with complex unit cell, the misfit cobaltates and the rare earth filled skutterudites.

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