Electronic frustration on a triangular lattice
N. P. Ong and R. J. Cava
Princeton University, Princeton, NJ, U.S.A.

Imagine you are hosting a dinner with an odd number of people in the party.  You quickly realize that it is impossible to seat every guest between two members of the opposite sex.  This is hardly a calamity, of course.  However, for electrons living on a triangular lattice, the concept of “geometrical frustration” in the face of strong Coulomb mutual repulsion is a crucial, decisive factor that shapes their behavior (Fig. 1a).  In an insulating lattice, the Coulomb repulsion force is relieved if each electron can point its spin antiparallel to that of its nearest neighbors.  On most lattices, this is readily implemented and leads to the Néel state in which spins alternate up and down along each bond direction.  On the triangular lattice, however, geometric frustration precludes such ideal regularity.  As implied in Fig. 1a, 2 of the 3 electrons in each triangle must share the same spin orientation.  At zero Kelvin, the spins remain in a disordered quantum state with no discernible pattern (often called a “spin liquid”).  Understanding the spin-liquid state is a major goal of the science of strongly-correlated materials (these are materials in which the Coulomb force is very large compared to familiar metals like gold).  Further, the problem is greatly enriched if the electrons are free to hop between sites and carry an electrical current.  Does electron itinerancy relieve the geometrical frustration?  Does the disordered quantum spin state leave its imprint on the conductivity?  Can these electrons form Cooper pairs to produce superconductivity?

These theoretical questions are relevant to recent experiments (1-5) on the cobalt oxide NaxCoO2 in which the Co ions define a layered triangular lattice (the Na ions are sandwiched between CoO2 layers).  Initial results (1) obtained on as-grown materials with x close to 2/3 showed that the material is an excellent, if unremarkable, electrical conductor.  However, its magnetic susceptibility is decidedly odd (the susceptibility measures how well a magnetic field aligns the electron spins).  In a metal, the fraction of electron spins that can be field-aligned is very small, and steadily shrinks to zero in decreasing temperature.  By contrast, in NaxCoO2, the susceptible spin population just equals the charge carrier population (holes), and stays unchanged with falling temperature (2,3).  The susceptibility resembles that of an insulator that is frustrated from attaining the ordered Néel state as described above.  This Janus-like ambivalence (metallic in charge conduction but insulator-like in spin alignment) has been dubbed a “Curie-Weiss metal” (4).

Equally puzzling in the initial study (1) was the finding that NaxCoO2 has an enhanced thermopower.  In metals, an electrical (or charge) current involves the flow of electrons, but because electrons carry entropy, the charge current is accompanied by an electronic heat current.  The thermopower measures the ratio of the heat to the charge current.  As a rule, the thermopower in metals is very small because, in an electric field, nearly equal populations of electrons and holes flow in opposite directions (holes are carriers with positive charge).  A recent experiment (3) has unearthed a vital clue to the large thermopower in NaxCoO2.  At 2 Kelvin, a magnetic field completely suppresses the thermopower to zero (this is possibly the first such observation in any solid).  Quantitative estimates confirm that the vanishing of the thermopower coincides with the complete alignment of the spins by the field (3).  By inference, this implies that the thermopower derives mostly from the spin entropy carried by the holes in the Curie-Weiss phase.

A yet bigger surprise emerged when it was found that water molecules intercalate readily to form a layer between the Na ions and the CoO2 layer.  If the water-logged sample NaxCoO2.yH2O is cooled below 4 Kelvin, it becomes a superconductor (5).  The superconducting phase is restricted to the narrow interval ¼ < x < 1/3 (6).  The essential role of water molecules in stabilizing the superconducting state is being intensely investigated (6-9).  An important factor seems to be the ability of water molecules to screen the strongly fluctuating electrostatic potential of the Na ions from the charge carriers in the CoO2 layers.  A second question is whether the pairing symmetry is unconventional (as in the cuprates).  However, measurements of the superconducting properties are greatly hampered by the difficulty of growing single crystals with the optimal water content: preparing crystals by electrochemical deintercalation may be the key (8).

As noted, the hole density in NaxCoO2 may be increased by reducing the Na content x.  In principle, with maximum Na content (x = 1), there are no holes on the lattice.  As the Na content is reduced, the holes increase in proportion, until every lattice site is occupied at x = 0.  Determining how the electronic properties vary with hole concentration is essential for understanding this material.  Recent progress has allowed the phase diagram to be established in the range ¼ < x < ¾ (Fig. 1b) (5).  As known form earlier work (3, 6), the interval x <0.4 includes the superconducting phase (with water intercalation), while the region near x = 2/3 harbors the Curie-Weiss metal described above.  The different hole densities in the 2 phases are confirmed by Fermi Surface measurements using photoemission spectroscopy (10).  Do these 2 distinct phases evolve smoothly into each other?   The phase diagram reveals that the answer is no.  Unexpectedly, a new state, occupying a strip centered at x = ½, rises like a firewall between them.  In this state, the material is a very poor electrical conductor.  Apparently, with half of the sites occupied by holes, the system has found a new way to accommodate both the strong Coulomb forces and geometric frustration by firmly localizing the holes (so they lose their ability to carry a current).  The mechanism for the formation of this “charge-ordered” insulating phase is currently an open issue.  Finally, for x > ¾, there are hints that the material attains very weak magnetic ordering at low temperatures.

Since the discovery of cuprate superconductivity in 1986, researchers have increasingly turned to novel materials ( notably manganates, ruthenates, and nickelates ) in which strong electron-electron interactions prevail, to explore new realms of condensed matter physics.  As in NaxCoO2, the array of phenomena uncovered presents a major challenge to conventional ideas.  Nonetheless, steady progress has occurred in the difficult task of incorporating strong electron interactions into the quantum theory of solids.

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