## Harris, A. Brooks

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Publication Landau Analysis of the Symmetry of the Magnetic Structure and Magnetoelectric Interaction in Multiferroics(2007-08-28) Harris, A. BrooksThis paper presents a detailed instruction manual for constructing the Landau expansion for magnetoelectric coupling in incommensurate ferroelectric magnets, including Ni3V2O8, TbMnO3, MnWO4, TbMn2O5, YMn2O5, CuFeO2, and RbFe(MO4)2. The first step is to describe the magnetic ordering in terms of symmetry adapted coordinates which serve as complex-valued magnetic order parameters whose transformation properties are displayed. In so doing, we use the previously proposed technique to exploit inversion symmetry, since this symmetry has seemingly been universally overlooked. Inversion symmetry severely reduces the number of fitting parameters needed to describe the spin structure, usually by fixing the relative phases of the complex fitting parameters. By introducing order parameters of known symmetry to describe the magnetic ordering, we are able to construct the trilinear magnetoelectric interaction which couples incommensurate magnetic order to the uniform polarization, and thereby we treat many of the multiferroic systems so far investigated. In most cases, the symmetry of the magnetoelectric interaction determines the direction of the magnetically induced spontaneous polarization. We use the Landau description of the magnetoelectric phase transition to discuss the qualitative behavior of various susceptibilities near the phase transition. The consequences of symmetry for optical properties such as polarization induced mixing of Raman and infrared phonons and electromagnons are analyzed. The implication of this theory for microscopic models is discussed.Publication Multiple Species of Noninteracting Molecules Adsorbed on a Bethe Lattice(2008-10-15) Cohen, Michael; Harris, A. BrooksA simple method, previously used to calculate the equilibrium concentration of dimers adsorbed on a Bethe lattice as a function of the dimer activity, is generalized to solve the problem of a Bethe lattice in contact with a reservoir containing a mixture of molecules. The molecules may have arbitrary sizes and shapes consistent with the geometry of the lattice and the molecules do not interact with one another except for the hard-core restriction that two molecules cannot touch the same site. We obtain a set of simultaneous nonlinear equations, one equation for each species of molecule, which determines the equilibrium concentration of each type of molecule as a function of the (arbitrary) activities of the various species. Surprisingly, regardless of the number of species, the equilibrium concentrations are given explicitly in terms of the solution of a single equation in one unknown which can be solved numerically, if need be. Some numerical examples show that increasing the activity of one species need not necessarily decrease the equilibrium concentration of all other species. We also calculate the adsorption isotherm of an “annealed” Bethe lattice consisting of two types of sites which differently influence the activity of an adsorbed molecule. We prove that if the reservoir contains a finite number of molecular species, regions of two different polymer densities cannot simultaneously exist on the lattice. The widely used Guggenheim theory of mixtures, which can also be construed as a theory of adsorption, assumes for simplicity that the molecules in the mixture are composed of elementary units, which occupy sites of a lattice of coordination number q. Guggenheim’s analysis relies on approximate combinatorial formulas which become exact on a Bethe lattice of the same coordination number, as we show in an appendix. Our analysis involves no combinatorics and relies only on recognizing the statistical independence of certain quantities. Despite the nominal equivalence of the two approaches, the easily visualized properties of the Bethe lattice enable one to solve some apparently difficult problems by quite elementary methods.Publication SpinWaves in the Frustrated Kagomé Lattice Antiferromagnet KFe3(OH)6(SO4)2(2006-06-19) Matan, Kittiwit; Grohol, Daniel; Yildirim, Taner; Nocera, Daniel G.; Harris, A. Brooks; Lee, Seunghun H.; Nagler, Stephen E.; Lee, Young SThe spin wave excitations of the S = 5/2 kagomé lattice antiferromagnet KFe3(OH)6(SO4)2 have been measured using high-resolution inelastic neutron scattering. We directly observe a flat mode which corresponds to a lifted ‘‘zero energy mode,’’ verifying a fundamental prediction for the kagomé lattice. A simple Heisenberg spin Hamiltonian provides an excellent fit to our spin wave data. The antisymmetric Dzyaloshinskii-Moriya interaction is the primary source of anisotropy and explains the low-temperature magnetization and spin structure.Publication Dimer Statistics on a Bethe Lattice(2006-11-13) Harris, A. Brooks; Cohen, MichaelWe discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice.We reproduce the well-known exact result for noninteracting hard-core dimers by both a very simple geometrical argument and a general algebraic formulation for lattice statistical problems. The algebraic formulation enables us to discuss loop corrections for finite dimensional lattices. For the Bethe lattice we also obtain the exact solution when either (a) the dimers interact via a short-range interaction or (b) the underlying lattice is anisotropic. We give the exact solution for a special limit of dimers on a Bethe lattice in a quenched random potential in which we consider the maximal covering of dimers on random clusters at site occupation probability p. Surprisingly the partition function for “maximal coverage” on the Bethe lattice is identical to that for the statistics of branched polymers when the activity for a monomer unit is set equal to −p. Finally we give an exact solution for the number of residual vacancies when hard-core dimers are randomly deposited on a one dimensional lattice.Publication Symmetry Analysis of Multiferroic Co3TeO6(2012-03-12) Harris, A. BrooksA phenomenological explanation of the magnetoelectric behavior of Co3TeO6 is developed. We explain the second harmonic generation data and the magnetic field induced spontaneous electric polarization in the magnetically ordered phase below 20 K.Publication Landau Theory of Tilting of Oxygen Octahedra in Perovskites(2012-05-18) Harris, A. BrooksThe list of possible commensurate phases obtained from the parent tetragonal phase of Ruddlesden-Popper (RP) systems, An+1BnC3n+1 for general n due to a single phase transition involving the reorienting of octahedra of C (oxygen) ions is reexamined using a Landau expansion. This expansion allows for the nonlinearity of the octahedral rotations and the rotation-strain coupling. It is found that most structures allowed by symmetry are inconsistent with the constraint of rigid octahedra, which dictates the form of the quartic terms in the Landau free energy. For A2BC4 our analysis allows only 10 of the 41 structures which satisfy the general symmetry arguments of Hatch et al. [Phys. Rev. B 39, 9282 (1989)]. The symmetry of rotations for RP systems with n > 2 is clarified. Our list of possible structures for general n excludes many structures allowed in previous studies.Publication A System Exhibiting Toroidal Order(2010-11-01) Harris, A. BrooksThis paper treats the dipolar interactions of a two-dimensional system of discs upon which a triangle of spins is mounted. We obtain the leading term of the multipole expansion of the interaction energy of discs on which is mounted a regular n-gon of spins. A definition of the toroidal magnetic moment Ti of the ith plaquette is proposed such that the magnetostatic interaction between plaquettes i and j is proportional to TiTj. The system for n=3 is shown to undergo a sequence of interesting phase transitions as the temperature is lowered. We are mainly concerned with the “solid” phase in which bond-orientational order but not positional order is long ranged. As the temperature is lowered in the solid phase, the first phase transition involving the orientation or toroidal magnetism of the discs is into a “gauge toroid” phase in which the product of a magnetic toroidal parameter and an orientation variable (for the discs) orders but due to a local gauge symmetry these variables themselves do not individually order. Finally, in the lowest temperature phase the gauge symmetry is broken and toroidal order and orientational order both develop. In the “gauge toroidal” phase time-reversal invariance is broken and in the lowest temperature phase inversion symmetry is also broken. In none of these phases is there long-range order in any Fourier component of the average spin. Symmetry considerations are used to construct the magnetoelectric free energy and thereby to deduce which coefficients of the linear magnetoelectric tensor are allowed to be nonzero. In none of the phases does symmetry permit a spontaneous polarization.Publication Theoretical Analysis of the Double-q Magnetic Structure of CeAl2(2006-10-17) Harris, A. Brooks; Schweizer, J.A model involving competing short-range isotropic Heisenberg interactions is developed to explain the double-q magnetic structure of CeAl2Θ. For suitably chosen interactions, terms in the Landau expansion quadratic in the order parameters explain the condensation of incommensurate order at wave vectors in the star of (1/2− δ,1/2+δ,1/2)(2π/a), where a is the cubic lattice constant. We show that the fourth-order terms in the Landau expansion lead to the formation of the so-called double-q magnetic structure in which long-range order develops simultaneously at two symmetry-related wave vectors, in striking agreement with the magnetic structure determinations. Based on the value of the ordering temperature and of the Curie-Weiss temperature Θ of the susceptibility, we estimate that the nearest-neighbor interaction K0 is ferromagnetic with K0/k=−11±1 K and the next-nearest neighbor interaction J is antiferromagnetic with J/k=6±2 K. We also briefly comment on the analogous phenomena seen in the similar system TmS.Publication Effect of Inversion Symmetry on the Incommensurate Order in Multiferoic RMn2O5 (R = Rare Earth)(2008-07-03) Harris, A. Brooks; Kenzelmann, M.; Aharony, Amnon; Entin-Wohlman, OraStarting from the irreducible representations of the group of the wave vector, we construct the spin-wave functions consistent with inversion symmetry, neglected in the usual representation analysis. We obtain the relation between the basis functions of different members of the star of the wave vector. We introduce order parameters and determine their transformation properties under the operations of the space group of the paramagnetic crystal. The results are applied to construct terms in the magnetoelectric interaction, which are quadratic and quartic in the magnetic order parameters. The higher-order magnetoelectric interactions can in principle induce components of the spontaneous polarization, which are not allowed by the lowest-order magnetoelectric interaction. We also obtain the relation between the spin-wave functions of the incommensurate phase and those of the commensurate phase, which lead to analogous relations between the order parameters of these two phases.Publication Symmetry analysis for the Ruddlesden-Popper systems Ca3Mn2O7 and Ca3Ti2O7(2011-08-24) Harris, A. BrooksWe perform a symmetry analysis of the zero-temperature instabilities of the tetragonal phase of Ca3Mn2O7 and Ca3Ti2O7 which is stable at high temperature.We introduce order parameters to characterize each of the possible lattice distortions to construct a Landau free energy which elucidates the proposed group-subgroup relations for structural transitions in these systems. We include the coupling between the unstable distortion modes and the macroscopic strain tensor. We also analyze the symmetry of the dominantly antiferromagnetic ordering which allows weak ferromagnetism. We show that in this phase the weak ferromagnetic moment and the spontaneous ferroelectric polarization are coupled, so that by rotating one of these orderings by applying an external electric or magnetic field one can rotate the other ordering.We discuss the number of different domains (including phase domains) which exist in each of the phases and indicate how these may be observed. First-principles calculations of Yildirim corroborate our assertion that domain walls in the nonferroelectric phase are narrow.