ARTICLES PUBLISHED ONLINE: 31 JULY 2011 | DOI: 10.1038/NPHYS2045 Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions Stefan Heinze 1 * † , Kirsten von Bergmann 2 * † , Matthias Menzel 2† , Jens Brede 2 , André Kubetzka 2 , Roland Wiesendanger 2 , Gustav Bihlmayer 3† and Stefan Blügel 3 Skyrmions are topologically protected ﬁeld conﬁgurations with particle-like properties that play an important role in various ﬁelds of science. Recently, skyrmions have been observed to be stabilized by an external magnetic ﬁeld in bulk magnets. Here, we describe a two-dimensional square lattice of skyrmions on the atomic length scale as the magnetic ground state of a hexagonal Fe ﬁlm of one-atomic-layer thickness on the Ir(111) surface. Using spin-polarized scanning tunnelling microscopy we can directly image this non-collinear spin texture in real space on the atomic scale and demonstrate that it is incommensurate to the underlying atomic lattice. With the aid of ﬁrst-principles calculations, we develop a spin model on a discrete lattice that identiﬁes the interplay of Heisenberg exchange, the four-spin and the Dzyaloshinskii–Moriya interaction as the microscopic origin of this magnetic state. A bout 50 years ago Skyrme 1 realized that topologically protected defects in continuous fields, now called skyrmions, share essential properties with single particles; for example, they are localized in space, have quantized topological charges, are subject to attractive or repulsive interactions, assemble in ordered phases and undergo phase transitions. Since this seminal paper, skyrmions have developed into a general concept in physics ranging from elementary particles 1 to liquid crystals 2 , Bose–Einstein condensates 3 and quantum Hall magnets 4,5 , and show similarities to the vortex lattice in type II superconductors 6 . They have been predicted to exist also in magnets 7 , and were recently explored in the bulk magnets MnSi (ref. 8), FeCoSi (ref. 9) and FeGe (ref. 10) with a chiral crystal structure using neutron scattering and Lorentz transmission electron microscopy. In these systems skyrmions with diameters of about 20–90 nm were induced by an external magnetic field out of a helical magnetic ground state. A spontaneous skyrmion-like phase was reported in MnSi in the vicinity of the helical transition temperature 11 , however, the existence of a skyrmion lattice as the spontaneous ground state of a magnet has so far only been proposed theoretically on the basis of a phenomenological field model 12 . A key ingredient in the theories of magnetic skyrmions is the Dzyaloshinskii–Moriya (DM) interaction, which has only recently been discovered to play a significant role for magnetic nanostructures at surfaces 13,14 . It is due to spin–orbit coupling, which links the spin to the real space, and it can occur in all systems with broken structure-inversion symmetry such as surfaces. As the DM interaction induces spin spirals of one particular rotational sense it is often denoted as a chiral interaction that is essential for the formation of magnetic skyrmions 12,15,16 . Up to now only DM-induced spin spirals propagating along one direction have been observed at surfaces 13,14 . Skyrmions can be viewed as their two- dimensional analogons; however, a possible mechanism stabilizing them against decay into one-dimensional spirals in the absence of an external magnetic field is not firmly established. It had been speculated before 17 that the two-dimensional nanoscale magnetic 1 Institute of Theoretical Physics and Astrophysics, University of Kiel, 24098 Kiel, Germany, 2 Institute of Applied Physics, University of Hamburg, 20355 Hamburg, Germany, 3 Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany. † These authors contributed equally to this work. *e-mail: heinze@physik.uni-kiel.de; kbergman@physnet.uni-hamburg.de. structure of an Fe monolayer (ML) on Ir(111) (ref. 18) may be a candidate for such a two-dimensional spin texture. Here, we present a complementary experimental and theoretical study revealing the existence of a skyrmion lattice as the spontaneous ground state of this magnetic system. Following the experimentally determined size of the magnetic unit cell of about 1 nm by 1 nm, we propose a magnetic state as shown in Fig. 1a, which is a lattice of atomic-scale skyrmions. The skyrmion number—a topological index of the field configuration—is defined by S = 1 4π  n·  ∂ n ∂ x × ∂ n ∂ y  dx dy (1) where n is the unit vector of the local magnetization and the integral is taken over the two-dimensional unit cell. The integrand of equation (1) defines a topological magnetic field that gives rise to a Lorentz force and the skyrmion number is proportional to the accumulated Berry phase in real space of travelling electrons in a skyrmion lattice 19 . For any trivial magnetic structure such as a ferro- or antiferromagnet S = 0 holds. In contrast, the spin texture shown in Fig. 1a is built from single magnetic units with a non-vanishing positive value of S (see Supplementary Section S1 for the skyrmion density), revealing the skyrmionic character of this topologically protected magnetic state. It resembles the skyrmion lattice proposed in ref. 12, but with the key difference that it occurs on the atomic length scale and is not a modulation of an underlying magnetic order, but is formed directly from the localized moments of the iron atoms. Therefore, it requires a theory that takes the discrete atomic lattice into account, to which the concept of skyrmions can also be extended 20,21 . Owing to its size we denote it as a nanoskyrmion lattice in the following. Real-space measurement Spin-polarized scanning tunnelling microscopy (SP-STM) is applied as it enables us to resolve magnetic structures in real space down to the atomic scale 22 . The Fe ML grows pseudomorphically NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics 1 © 20 11 M acmillan Publishers Limited. All rights reserved.