Session W8: Focus Session: Frustrated Magnetism - Pyrochlore lattice

Topological phases in R2Ir2O7

We construct and analyze a theoretical model for the pyrochlore iridates R$_2$Ir$_2$O$_7$ with R (= Pr, Nd, Sm, Gd, Tb, Dy, Ho, Yb) magnetic. The electrons on trivalent rare earth ions R$^{3+}$ form local Ising doublets due to the local crystal field. Based on a space group symmetry analysis, we write down the generic Kondo coupling between the Ising spin at R sites and the effective spin at Ir sites. Besides this interaction, we also include direct electron tunneling between Ir sites and indirect electron tunneling via intermediate oxygens for Ir-Ir coupling. This simple minimal model gives a rich phase diagram with broad regions of topological semi-metal and axion insulator phases. Based on these findings, we propose R$_2$Ir$_2$O$_7$ to be one of the most promising candidates to realize the topological semi-metal and axion insulator phases. Implications for existing and future experiments are discussed.   

Muon Spin Relaxation Studies of Pyrochlore Iridates R$_2$Ir$_2$O$_7$, (R= Y, Yb, and Nd)

We report results for zero field muon spin relaxation measurements over the range 2 K $<$ T $<$ 220 K taken at the ISIS facility for three members of the pyrochlore iridate family, R$_2$Ir$_2$O$_7$, comprised of both magnetic (R = Yb, Nd) and non-magnetic (R = Y) species. The formation of a static internal field is detected via a loss of asymmetry below the bifurcation temperature observed in magnetic susceptibility for all three compounds. We will present results describing the nature of this internal field based on short time muon depolarization data taken at PSI. Longitudinal field measurements reveal the internal field is of order 1000 G for all three materials, and the existence of dynamic fluctuations at 2 K for R = Yb, Nd; the depolarization is quasistatic near 2 K for R = Y. We discuss these results as applied to understanding the unusual magnetic and transport properties observed in the iridate family, and the implications regarding long range magnetic order and the low temperature ground state.   

Chiral RKKY interaction in Pr2Ir2O7

Pr$_2$Ir$_2$O$_7$ experimentally realizes a chiral spin liquid selected out of a degenerate quantum spin-ice manifold. We propose that such a chiral state is selected by an analogue of the magnetic RKKY effect, whereby the chiral fluctuations of conducting Ir electrons close to a Mott transition promote correlations between the chirality of the local Pr moments. Pr$_2$Ir$_2$O$_7$ provides the perfect setting for such a \emph{chiral RKKY} effect, as its spin ice manifold naturally contains chiral states, and chiral fluctuations in the Ir are enhanced by the proximity to the metal insulator transition between Pr$_2$Ir$_2$O$_7$ and Nd$_2$Ir$_2$O$_7$. We calculate the chiral susceptibility within a simplified model, showing that this chiral RKKY coupling can be ferro-chiral and estimating the magnitude.   

Emergent spin superstructures and degeneracy in models of itinerant magnets

In recent years, there has been immense research interest in frustrated magnets with metallic character, such as the pyrochlores R$_2$Mo$_2$O$_7$, where R denotes a rare-earth element (Science {\bf 291}, 2573 (2001)). The frustration in magnetic sector in such systems can have interesting consequences for the electronic properties. More interestingly, the electronic degree of freedom can aid the system in selecting unconventional magnetic states. Motivated by these possibilities we investigate the double-exchange model with competing super-exchange interactions on various lattices. On a triangular lattice we find a novel chiral spin state to be the ground state at half filling (PRL {\bf 105}, 216405 (2010)). On checkerboard and Kagome lattices, the itinerant electrons play a crucial role in lifting the degeneracy of the magnetic sector and select specific ground states with intriguing superstructures. A very interesting effect takes place on a honeycomb lattice, where a Yafet-Kittel type magnetic structure emerges from the interplay between the super-exchange and double-exchange interactions (PRL {\bf 107}, 076405 (2011)). Some of these emergent spin states have a macroscopic degeneracy related to a symmetry which is intermediate between local and global.   

Resolving magnetic frustration in a Laves lattice

CeFe2 is a ferromagnet that exhibits antiferromagnetic fluctuations in its ground state. We use x-ray diffraction and diamond-anvil-cell techniques to directly measure the transition to antiferromagnetism in pure CeFe2 at high pressure which couples to the change in the lattice symmetry. Numerical simulations are adopted to identify the magnetic structure of the ground states and to quantitatively illustrate effects of competing magnetic energy scales and geometrical frustration on the magnetic phase diagram. Comparison of phase transitions under both chemical substitution and applied pressure suggests a general solution to the physics of Laves phase magnets.   

Neutron Diffraction Investigation of Magnetic and Orbital Order in FeV$_{2}$O$_{4}$

The vanadium spinels, AV$_{2}$O$_{4}$, with divalent cations on the diamond sublattice are model magnetic systems for the study of interacting orbital, lattice and spin degrees of freedom. Studies of both systems with diamagnetic (e.g. Zn$^{2+}$, Cd$^{2+}$, Mg$^{2+})$ and spin-only (e.g. Mn$^{2+})$ cations on the A-site sublattice have revealed multiple phase transitions and ground state properties heavily influenced by V$^{3+}$ orbital degrees-of-freedom. I will report on neutron powder diffraction measurements of another spinel system, FeV$_{2}$O$_{4}$, which additionally has two-fold orbitally degenerate Fe$^{2+}$ cations on the A-site sublattice. Previous x-ray and Mossbauer studies have reported four structural phase transitions in this material and at least one magnetic transition. Our data confirm the existence of three structural transitions and reveal distortions of local polyhedra with important implications for orbital order. We confirm the existence of hypothesized collinear antiferromagnetism below a temperature T$_{N1}$=110K and further identify a second magnetic transition at T$_{N2}$=60K where V$^{3+}$ moments cant away from the Fe$^{2+}$ spin direction to form a 2-in-2-out spin structure on the pyrochlore sublatice. I will discuss these observations in the context of recent predictions for orbital order in vanadate spinels.   

Universal exchange-driven phonon splitting

We report on a linear dependence of the phonon splitting on the non-dominant exchange coupling $J_{nd}$ in the antiferromagnetic monoxides MnO, Fe$_{0.92}$O, CoO and NiO, and in the highly frustrated antiferromagnetic spinels CdCr$_{2}$O$_{4}$, MgCr$_{2}$O$_{4}$ and ZnCr$_{2}$O$_{4}$. For the monoxides our results directly confirm the theoretical prediction of a predominantly exchange induced splitting of the zone-centre optical phonon [1,2]. We find the linear relation $\hbar\Delta\omega = \beta J_{nd} S^2$ with slope $\beta$ = 3.7. This relation also holds for a very different class of systems, namely the highly frustrated chromium spinels. Our finding suggests a universal dependence of the exchange-induced phonon splitting at the antiferromagnetic transition on the non-dominant exchange coupling [3].\\[4pt] [1] S. Massidda et al., Phys. Rev. Lett. 82, 430 (1999).\\[0pt] [2] W. Luo et al., Solid State Commun. 142, 504 (2007).\\[0pt] [3] Ch. Kant et al., arxiv:1109.4809.   

First-principles study of the structural and magnetic phase transition in CdCr$_2$O$_4$

We use first-principles calculations to investigate the mechanism for the paramagnetic (PM) cubic to an antiferromagnetic (AFM) tetragonal structural phase transition at 7.8 K in the spinel oxide compound CdCr$_2$O$_4$. Because of the magnetic frustration associated with AFM interactions on the pyrochlore lattice formed by the Cr ions, we focus on the spin-lattice coupling, specifically (a) the forces on the atoms and resulting atomic displacements induced by the spin configuration and (b) the spin-phonon coupling. Using the linear response method, we determine the full phonon dispersion relations for the PM, FM and AFM orderings in cubic and tetragonal structures. We determine the strength of spin-phonon couplings using IFCs for various magnetic orderings and show that the spin-phonon couplings are large but do not lead to any unstable modes that could alone drive the structural transition at low temperatures. Instead, we find that it is the symmetry-lowering forces and stresses induced by AFM ordering that drive the cubic to tetragonal phase transition at low temperature. The results are compared with a recent experimental determination of the phonon dispersion in the in the cubic and tetragonal phases of CdCr$_2$O$_4$ and the implications for related compounds discussed.   

Frustration and Jahn-Teller ordering in magnetic spinel oxides

Geometrically frustrated magnetic systems have become extremely important for the study of novel phenomena often present in highly degenerate magnetic ground states. Extensive study of the canonically frustrated ZnCr$_2$O$_4$ and MgCr$_2$O$_4$ has increased the understanding of frustration in three-dimensional systems, however little is known about the structural effect of dilute magnetism on the $A$ site. We present the effects on magneto-structural coupling of dilute magnetic cations on the diamagnetic $A$ site of ZnCr$_2$O$_4$ and MgCr$_2$O$_4$. In ZnCr$_2$O$_4$ and MgCr$_2$O$_4$, a spin-driven structural distortion concomitant with the onset of long-range magnetic order resolves the large ground state degeneracy at temperatures far below the theoretical Curie-Weiss temperature. Employing variable temperature high-resolution synchrotron powder diffraction studies and magnetic susceptibility measurements, we show the changes in spin-lattice coupling in the solid solutions Zn$_{1-x}$Co$_x$Cr$_2$O$_4$, Zn$_{1-x}$Cu$_x$Cr$_2$O$_4$ and Mg$_{1-x}$Cu$_x$Cr$_2$O$_4$ for $x \leq 0.2$. These results highlight the effect of dilute $A$ site magnetism on magnetic frustration as well as the role of dilute Jahn-Teller active Cu$^{2+}$ on spin-lattice coupling in these frustrated systems.   

Ultrasound Velocity Measurements in the Geometrically Frustrated Spinel MgCr$_2$O$_4$

Magnesium chromite spinel MgCr$_2$O$_4$ is a geometrically frustrated magnet with the N$\acute{e}$el temperature $T_N\simeq$13 K, and the Weiss temperature $\theta_W = -390$ K. Recent inelastic neutron scattering experiments provided a compelling evidence for the spin molecular ground states in not only the paramagnetic phase but also the antiferromagnetic phase. We performed ultrasound velocity measurements of MgCr$_2$O$_4$ in all the symmetrically independent elastic moduli of $C_{11}$, $(C_{11}-C_{12})/2$, and $C_{44}$. Temperature dependence of all of these elastic moduli exhibits a remarkable softening in the paramagnetic phase. Taking into account the absence of orbital degrees of freedom in Cr$^{3+}$ (3$d^3$) in MgCr$_2$O$_4$, the spin degrees of freedom should play a significant role for the elastic softening. The most probable origin for the elastic softening in the paramagnetic phase is the strong coupling of the acoustic phonons to the molecular spin fluctuations.   

Magnetic correlations and magnetic order in the pyrochlore Er$_2$Ti$_2$O$_7$

We analyse short-range magnetic correlations in the pyrochlore magnet Er$_2$Ti$_2$O$_7$. Four unique nearest-neighbour exchange interactions are permitted by the space group symmetry of the pyrochlore lattice; the four corresponding nearest-neighbour exchange constants for Er$_2$Ti$_2$O$_7$ are extracted from diffuse neutron scattering maps. Low-temperature magnetic order in Er$_2$Ti$_2$O$_7$ is discussed in light of these results. The results are compared to recently published values for the sister compound Yb$_2$Ti$_2$O$_7$, which has similar features.   

Low Temperature Spin Structure of Gadolinium Titanate

Many rare earth pyrochlore oxides exhibit exotic spin configurations at low temperatures due to frustration. The nearest neighbor coupling between spins on the corner-sharing tetrahedral network generate geometrical magnetic frustration. Among these materials, gadolinium titanate (Gd2Ti2O7) is of particular interest. Its low temperature ordered phases are not yet understood theoretically. Bulk thermal measurements such as specific heat and magnetic susceptibility measurements find two phase transitions in zero external field, in agreement with simple mean field calculations. However, recent neutron scattering experiments suggest a so-called 4-k spin structure for intermediate phase and a so called canted 4-k structure for lower temperature phase that does not agree with either mean-field theory or Monte Carlo simulation which find the 1-k state and Palmer-Chalker state respectively as the lowest free energy configuration for those phases. In our work, we study the 4-k structure in detail and present a new phase diagram for dipolar Heisenberg spins on a pyrochlore lattice, certain portions of which describe gadolinium titanate.   

Elastic properties of the titanate pyrochlore Tb$_{2}$Ti$_{2}$O$_{7}$

The presence of geometric frustration inhibits the formation of long-range spin ordering in Tb$_{2}$Ti$_{2}$O$_{7}$ even at very low temperatures, making this compound the prototype of a ``spin liquid.'' We have initiated a study of the elastic properties of Tb$_{2}$Ti$_{2}$O$_{7}$ as a function of temperature (0.5-300 K) and magnetic field (0-15T) using Resonant Ultrasound Spectroscopy (RUS). All three elastic constants show a pronounced softening below 50 K, indicative of a possible structural transition at very low temperatures. Application of a magnetic field partially suppresses the elastic softening in this compound, suggesting that the magnetoelastic coupling plays a significant role in the spin liquid physics of Tb$_{2}$Ti$_{2}$O$_{7}$ at low temperatures.   

Imaging Magnetic Order in Magnetostructural Phases of Mn3O4

Frustration in A-site spinels due to the competition between complex structure and strong interactions has been the focus of many theoretical and experimental studies recently. Mn3O4 is one such material with a three way interplay between complex lattice geometry, strong spin-lattice coupling and magnetic interactions. Mn3O4 is known to have two distinct phases below the Neel temperature that differ in both structure and magnetic order, including a tetragonal phase with disordered spins, and at lower temperatures, an orthorhombic phase exhibiting long-range commensurate magnetic order. In this talk, we present a nanometer-scale imaging investigation of the transition between these two phases. With cryogenic magnetic force microscopy (MFM), we observe novel magnetic stripe patterns accompanying this phase transition. Complementary electron backscatter diffraction (EBSD) measurements indicate that the magnetic patterns are closely related to local crystalline orientation. The onset and spatial periodicity of the magnetic patterns show variations with temperature and external magnetic field. We will also discuss possible causes of this phenomenon and their implications.   

Spin state of Mn$_{3}$O$_{4}$ investigated by $^{55}$Mn NMR

The $^{55}$Mn nuclear magnetic resonance spectrum for spinel oxide Mn$_{3}$O$_{4}$ was measured in the temperature range of 6 K - 30 K to investigate the spin structure in the ground state. The spectrum consists of three peaks in the frequency range of 250 - 265 MHz, which corresponds to the hyperfine field range of 24 - 25 T. The temperature dependence of the spectrum and the rf enhancement factor show that Mn$^{3+}$ ions have two different magnetic moments, one of which is strongly related with the commensurate-incommensurate phase transition. This is consistent with the picture of two magnetic moments, R and S, claimed from the result of a neutron experiment. The dipolar hyperfine field was calculated to explain the splitting of two peaks coming from R and S and to estimate the magnetic moments. The spin-spin relaxation time has a frequency dependence that induces spectrum broadening and further splitting of the peak coming from S, indicating that the Suhl-Nakamura interaction is the major relaxation mechanism in Mn$_{3}$O$_{4}$.