Electronic Supplementary Material (ESI) for Physical Chemical Physics. This journal is © the Owner Societies 2020

Tetragonal phase of 푀푛2퐵2 sheet: A stable room temperature ferromagnet with sizable magnetic anisotropy

ae, b ac b Yusuf Zuntu Abdullahi Zeynep Demir Vatansever , Ethem Aktürk , Ümit Akıncı , Olcay Üzengi Aktürkacd

a Department of Physics, Adnan Menderes University, Aydın 09010, Turkey. b Faculty of Science, Physics Department, Dokuz Eylül University, Tınaztepe Campus, 35390 İzmir, Turkey. cNanotechnology Application and Research Center, Adnan Menderes University, Aydın 09010, Turkey dDepartment of Electrical and Electronic Engineering, Adnan Menderes University, 09100 Aydın, Turkey eDepartment of Physics, Faculty of Science, State University, P.M.B. 2339 , .

Table of Contents 1. Linear response approach for 퐻푢푏푏푎푟푑 푈 parameter calculation

2. Snapshots of MD at 2 ps for tetr-푀푛2퐵2 sheet 3. Three different magnetic configurations 4. Spin configuration for the exchange-interaction constants 5. Supporting Tables

1. Linear response approach for 퐻푢푏푏푎푟푑 푈 parameter calculation:

Linear response approach formulated by Cococcioni et al. [1] was used to calculate the 퐻푢푏푏푎푟푑 푈 parameter of the 푀푛 atom in the . The 퐻푢푏푏푎푟푑 푈 parameter is extracted from density response function using the following equation:

2 ∂ 퐸 ∂푛푝 휘푝푞 = = ∂훼푝훼푞 ∂훼푞 (1)

푛푝represents the occupation of d orbital localized states with respect to site 푝 and 훼 variable is the perturbation potential. Once the density response function is known, the total on-site effective 푈 value is self-consistently determined, and is expressed as the equation below, ‒ 1 ‒ 1 푈푒푓푓 = 휘 0 ‒ 휘 (2)

휘 and 휘0 denote the interacting and non-interacting density response of the system with respect to localized perturbations. The obtained Hubbard 푈 for 푀푛 atom used in this work is shown in Fig. S1. The 퐻푢푏푏푎푟푑 푈 value calculated in this study is lower than that adopted by Jian et al. [2]. The discrepancy is the result of the differences in the bonding environments being considered.

Figure S1: Linear response of 푑 orbital occupations as a function of potential shift 훼. The curves depicted by the squares and circles lines are labeled bare and interacting. The inverse response functions are deduced numerically by calculating the slope of the curves. 휘0follows from the slope of curve bare, whereas 휘 from the slope of curve interacting. 푀푛 퐵 2. Snapshots of MD at 2ps for tetr- 2 2 sheet Figure S2: Top and side view 3× 3× 1 supercell structures of tetr-푀푛2퐵2 sheet under a Nose−Hoover thermostat at 300K, 600K and 900K, respectively. 3. Three different magnetic configurations

Figure S3. Top views (a−c) of geometric structures of an FM and two different AFM states for tetragonal 푀푛2퐵2 sheet. Red and blue color denote the spin up and spin down respectively.

3. Spin configuration for the exchange-interaction constants Figure S4. Slanted top view of spin configuration for estimating the exchange-interaction constants for tetragonal 푀푛2퐵2 sheet. d1 and d2 denote the first and second nearest neighbor respectively.

Supporting Tables

Table S1: Relative energies (in meV per unit cell) between FM and AFM. FM AFM1 AFM2 푈 = 0 푒푉 0 753.2 516.1 푈 = 3.32 푒푉 0 680 37.6

Table S2: Calculated J parameters in based on Heisenberg model and magnetic anisotropy energy (MAE) in 휇푒푉 (Per unit cell) for tetragonal 푀푛2퐵2 sheet. 푀푛2퐵2 sheet has FM as ground state for both cases. 푈 = 0 푒푉 푈 = 3.32 푒푉

3.231 4 . 342

15.180 0 . 739

퐸(100) ‒ 퐸(001) 870 1740 퐸(010) ‒ 퐸(001) 870 1740

퐸(111) ‒ 퐸(001) 570 1160

Table S3: Summary of Magnetic Anisotropy Energies (MAE) in 휇푒푉 per Mn atom. The electric field is applied along the 푧-direction. For all the cases, the easy axis is the out-of-plane [001] (perpendicular) direction. Electric field (eV/nm) 퐸(100) ‒ 퐸(001) 퐸(010) ‒ 퐸(001) 퐸(111) ‒ 퐸(001) 0 217.5 217.5 145 2 248.8 248.8 166.3 4 250 250 167.5

Reference [1] M. Cococcioni, S. De Gironcoli, Phys. Rev. B., 71 (2005) 035105. [2] Z. Jiang, P. Wang, X. Jiang, J. Zhao. Nanoscale Horizons. 2018 3 (3) 33541.