Magnetic frustration in an iron based Cairo pentagonal lattice E. Ressouche, 1,$_{∗}$ V. Simonet, 2 B. Canals, 2 M. Gospodinov, 3 and V. Skumryev 4 $^{1}$CEA/Grenoble, INAC/SPSMS-MDN, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France $^{2}$Institut N´ eel, CNRS & Universit´ e Joseph Fourier, BP166, 38042 Grenoble Cedex 9, France $^{3}$Institute of Solid State Physics, Bulgarian Academy of Sciences, 1184 Sofia, Bulgaria $^{4}$Instituci´ o Catalana de Recerca i Estudis Avanats (ICREA) and Departament de F´ısica, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Spain (Dated: October 30, 2018) The Fe 3+ lattice in the Bi 2 Fe 4 O 9 compound is found to materialize the first analogue of a magnetic pentagonal lattice. Due to its odd number of bonds per elemental brick, this lattice, subject to first neighbor antiferromagnetic interactions, is prone to geometric frustration. The Bi 2 Fe 4 O 9 magnetic properties have been investigated by macroscopic magnetic measurements and neutron diffraction. The observed non-collinear magnetic arrangement is related to the one stabilized on a perfect tiling as obtained from a mean field analysis with direct space magnetic configurations calculations. The peculiarity of this structure arises from the complex connectivity of the pentagonal lattice, a novel feature compared to the well-known case of triangle-based lattices. The pentagon, a 5-edges polygon, is an old issue in mathematical recreation. It forms the faces of the do- decahedron, one of the platonic solids whose shape is re- produced in biological viruses and in some metallic clus- ters [1]. The main peculiarity of this polygon is that, contrary to triangles, squares or hexagons, it is impossi- ble to tile a plane with congruent regular pentagons, the tilings must involve additional shapes to fill the gaps [2], like in the Penrose lattice (Fig. 1). It exists however several possibilities of tessellation of a plane with non- regular pentagons, a famous one being the Cairo tessel- lation whose name was given because it appears in the streets of Cairo and in many Islamic decorations (Fig. 1). Such a lattice could attract interest in the field of ge- ometric frustrated magnetism. Indeed, frustration usu- ally arises when all pairs of magnetic interactions are not simultaneously satisfied in a system due to the lat- Bi$_{2}$Fe$_{4}$O$_{9}$ is a common by-product in the synthesis of the multiferroic compound BiFeO$_{3}$, and has been claimed recently to display itself multiferroic properties [4]. This was the initial motivation of our study: this finding re- quires a precise determination of the symmetries in the magnetic ordered phase, since ferroelectricity can only arise in polar space groups. Such a precise determination has never been carried out on single crystals of Bi$_{2}$Fe$_{4}$O$_{9}$, and only powder neutron diffraction measurements have been reported [5] with no definitive conclusions due to the complexity of this orthorhombic structure. Actu- ally, there are two different sites of four iron atoms in Bi$_{2}$Fe$_{4}$O$_{9}$: Fe$_{1}$ occupies a tetrahedral position and Fe$_{2}$ an octahedral one. In the structure, columns of edge- sharing Fe$_{2}$ octahedra form chains along the c axis, and these chains are linked together by corner-sharing Fe$_{1}$ tetrahedra and Bi atoms (Fig. 2). As a result, the lattice formed by the different Fe 3+ magnetic atoms is quite re- markable. From a simple examination of the structure, five main magnetic superexchange interactions, J$_{1}$ to J$_{5}$ , can be identified (Fig. 2). The structure can be simply viewed as layers perpendicular to the c axis: the mag- netic coupling of these layers along the c direction in- tice topology, resulting in unusual physical properties [3]. Today, most of the studied systems are mainly based on triangles (or tetrahedra in three dimensions), and can be made perfect from both structural (regular polygons) and magnetic (equal first neighbor magnetic interactions) points of view. This is not the case with pentagonal lat- tices that must involve only non-regular pentagons to fill the space. The experimental realization of such a model system is thus a unique opportunity that could bring new interesting features in this field of magnetic frus- tration. In this letter we show that the Fe 3+ lattice in Bi$_{2}$Fe$_{4}$O$_{9}$ materializes the first analogue of a magnetic pentagonal lattice, and we interpret its peculiar non- collinear magnetic arrangement with respect to magnetic frustration. arXiv:1001.0710v1 [cond-mat.str-el] 5 Jan 2010 FIG. 1: (color online) Pentagonal lattices. (a) Cairo pen- tagonal tiling, (b) Penrose pentagonal tiling, (c) pentagonal lattice studied in refs. 6–8, (d) dodecahedron.