All-Electron DFT Simulations of Particle-Like Magnetic Objects

**Principal Investigator:**

Stefan Blügel

**Affiliation:**

Forschungszentrum Jülich

**Local Project ID:**

pn72qa

**HPC Platform used:**

SuperMUC-NG at LRZ

**Date published:**

Complex magnetic textures and localized particle-like structures on the nanometer scale such as chiral magnetic skyrmions (Fig. 1) with non-trivial topological properties are nowadays the most studied objects in the field of nanomagnetism. They offer the promise of new data storage and data processing technologies ranging from racetrack memories to . The theoretical and computational description of nanosized magnetic objects usually follows a typical multiscale approach. Ab-initio simulations mostly based on Density Functional Theory (DFT) are utilized to determine fundamental materials properties. Based on these properties simpler purely-magnetic models like the extended Heisenberg model are constructed and spin-dynamics simulations are employed to describe the final complex magnetic structures. While this approach has demonstrated its power and efficiency, it cannot describe fundamental changes in the electronic structure induced by the complex alignment of magnetic moments. In this project we extend the realm of DFT calculations for magnetic systems to significantly larger setups to tackle these challenges from the basic description provided by our ab-initio code FLEUR [1]. Since 2015 FLEUR is one of the flagship codes of the MaX-European Center of Excellence in HPC [2]

Our project aims at providing insight into the electronic properties of large and complex non-collinear magnetic structures by applying our state-of-the-art all-electron full-potential linearized augmented plane wave (FLAPW) DFT code FLEUR [3]. While the FLAPW method is a numerically complex and computationally heavy approach, it is also considered to provide the ”gold standard” and reference results in DFT. Its accuracy and reliability proven in numerous applications is particularly crucial in the field of magnetism as the fundamental energy scales on which magnetic phenomena occur is usually very small. While DFT simulations of large setups containing many atoms are already possible for quite some time, the simulations we pursue extend the limits of so called all-electron DFT applied to large magnetic configurations.

A DFT simulation is an iterative process with 50-120 cycles to achieve a self-consistent solution of the fundamental quantum mechanical equations. A single of these self-consistency iterations contains many different steps like the determination of the potential, the generation of a new density and the density mixing to accelerate the self-consistency process. From the computational point of view, the most relevant task in each iteration consists of the setup of matrices for the generalized eigenvalue problem and its diagonalization. The hybrid MPI/OpenMP parallelization of the FLEUR code allows to utilize machines like SuperMUC-NG [4].

To get insight into the details of the electronic structure which are not in reach of conventional multiscale approaches, we performed simulations with two magnetic setups which are larger than any production-level FLAPW calculations reported so far: i) a globule with two Bloch points in the MnGe 4x4x8 super cell (Fig. 2) and ii) a skyrmion in a HBi film (Fig. 1).

Structure | # atoms | Matrix size | Iteration cost |

Globule | 1,024 | 150k x 150k | 5,120 core-h |

Skyrmion | 192 | 200k x 200k | 11,670 core-h |

The setups containing the Bloch points run on 256 nodes with 48 cores for about 25 minutes per iteration. Hence, a single self-consistency cycle consumes about 5000 core-hours and a full self-consistency with 165 iterations 825000 core-hours. As we have to compare the different magnetic states for a full analysis of the system and to gain inside knowledge of the interplay between the magnetic and electronic structure of these configurations, we need to obtain self-consistency for several setups. These calculations involve the diagonalization of several complex matrices of size 150k x 150k per iteration. The corresponding data for the second system reported here are given in Table 1. Typical for a DFT calculation, our IO requirements are relatively modest and only few files are generated.

In continuous models, a Bloch point manifests itself as a singularity at which the magnetization vanishes. This already indicates that drastic changes in the electronic structure can be expected. When a system with a Bloch point is simulated with spin-dynamic methods, only the directions of the magnetization vectors are allowed to relax. Our simulations show that the magnetization of the atom most close to the Bloch point is significantly reduced in value (Fig.3).

It has been shown [3] that hydrogenated Bi monolayers can exhibit large topological bandgaps and manifest a quantum spin Hall effect. As the partially hydrogenated Bi monolayer is magnetic, the effect of magnetic order on the electronic structure can be significant. In particular the complex interplay between the topological protection in the quantum anomalous state and the magnetic order introduces a rich playground in which details of the electronic structure can be controlled and manipulated by the arrangement of magnetic moments in the system. While such effects have been simulated by means of simple tight-binding models, these models can only indicate the basic phenomenon but fail to grasp the details. We therefore plan to study large unit-cells (as on the Fig. 1 and larger) with non-trivial magnetic order imposed.

[2] “MaX-Materials Design at the Exascale, a European centre of excellence”, www.max- centre.eu.

[3] C.Niu, G.Bihlmayer, H.Zhang, D.Wortmann,S.Blügel and Y.Mokrousov, “Functionalized bismuth films: Giant gap quantum spin Hall and valley-polarized quantum anomalous Hall states”, Phys.Rew. B 91, 041303 (2015)

[4] U. Alekseeva, G. Michalicek, D. Wortmann and S. Blügel, “Hybrid Parallelization and Performance Optimization of the FLEUR Code: New Possibilities for All-Electron Density Functional Theory”, in: Springer International Publishing AG, part of Springer Nature 2018 M. Aldinucci et al. (Eds.): Euro-Par 2018, LNCS 11014, pp. 735–748, 2018.