Room Temperature Quantum Anomalous Hall Effect in Single-Layer

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials 2

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Room Temperature Quantum Anomalous Hall Effect in Single-Layer CrPS Pei Zhao, Yandong Ma, Hao Wang, Baibiao Huang, and Ying Dai J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b04888 • Publication Date (Web): 27 May 2019 Downloaded from http://pubs.acs.org on May 30, 2019

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The Journal of Physical Chemistry

Room Temperature Quantum Anomalous Hall Effect in Single-Layer CrP2S6 Pei Zhao, Yandong Ma*, Hao Wang, Baibiao Huang and Ying Dai* School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Shandanan Str. 27, Jinan 250100, People's Republic of China *Corresponding author: [email protected] (Y. M); [email protected] (Y. D)

ABSTRACT Quantum anomalous Hall (QAH) effect is a fascinating quantum phenomenon characterized by a nonzero Chern number defined in the bulk and chiral edge states in the boundary. Up to now, only one magnetically doped topological insulator, suffering from small bulk band gap, is confirmed to host QAH effect experimentally. Here, through first-principles calculations, we propose a novel QAH insulator in single-layer (SL) CrP2S6. The nontrivial topology in SL CrP2S6, identified with the nonzero Chern number (C = -1) and chiral edge states, harbors a nontrivial band gap of 53 meV. Meanwhile, using Monte Carlo simulations, the Curie temperature Tc for its ferromagnetic order is estimated to be 350 K, which is above the room temperature and comparable with most of the previously reported two-dimensional ferromagnetic semiconductors. Our findings thus present a feasible platform for achieving QAH effect at room temperature.

Keywords: Single-layer; ferromagnetic; ternary transition-metal thiophosphate; quantum anomalous Hall effect. 1

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1. INTRODUCTION Quantum anomalous Hall (QAH) effect is considered as a fantastic electronic behavior, which has been sparking tremendous interest over the past decades.1-8 Two essential ingredients for realizing the QAH effect are the ferromagnetic (FM) order and spinorbital coupling (SOC), which conspire to give rise to a global band gap in the bulk and gapless conducting edge states at its boundary. Its topologically nontrivial electronic property is characterized by a nonzero Chern number C in the momentum space, which is in accordance with the number edge states. They are robust against impurity perturbations because the chiral edge states of the QAH insulator only allow one spin species to flow unidirectionally. In this sense, the chiral edge state of the QAH insulator can sever as dissipationless transport channels to transmit and store information, which is of great significance for the development of next-generation low-energy spintronic devices. Until now, extensive efforts have been devoted to exploring new QAH insulators. The ordinary avenue is to transform the quantum spin Hall (QSH) insulators into QAH insulators via inducing exchange magnetic field (i.e., doping transition metal atoms or forming

heterostructures

with

ferromagnetic/antiferromagnetic

substrates).9-12

Following this way, silicene, germanene and stanene can be converted into QAH insulators.13-17 Besides, many intrinsic QAH insulators, including hexagonal organometallic frameworks, transitional metal halides and transitional metal oxides, are proposed as well.18-29 However, in most of the previously reported QAH insulators, the nontrivial band gap and estimated Curie temperature are so small to reach the requirement of applications at room temperature (300 K). As a consequence, despite those extensive efforts, so far, only the Cr or V doped (Bi, Sb)2Te3 systems are identified to exhibit QAH effects in experiment. Hence, it is crucially important to search for novel intrinsic QAH insulators with large nontrivial gap and high Curie temperature. In this work, we propose an experimentally feasible QAH insulator, SL CrP2S6. Employing first-principles calculations, the nontrivial topological state in SL CrP2S6 is firmly confirmed by a nonzero Chern number (C = -1) and chiral edge states. The nontrivial band gap is found to be 53 meV and the Curie temperature for its ferromagnetic order is estimated to be ~350 K. Therefore, the room-temperature QAH 2


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effect is expected to be observed in SL CrP2S6, providing a compelling candidate for practical applications. Besides, we also investigate the electronic properties of SL CoP2S6 and MnP2S6, which are found to be half-metal and FM semiconductor with Curie temperature of 200 K and 150 K, respectively. Their Curie temperatures are comparable with those of the previous proposed SL ferromagnetic systems,30-32 making them promising building blocks for future applications in electronics and spintronics. 2. METHODS First-principles calculations are performed using projector-augmented wave (PAW) scheme implemented in Vienna ab-initio simulation package (VASP).33,34 The generalized gradient approximation (GGA) as formulated by Predew-Burke-Ernzerhof (PBE) is used to describe the exchange and correlation interactions.35 The cut-off energy of 500 eV is chosen for the plane-wave expansion of wave functions. To avoid the interaction between adjacent layers, a vacuum space of about 18 Å perpendicular to 2D plane is adopted. The Brillouin zone (BZ) is sampled by using a 9×9×1 grid. The structures are fully relaxed until the residual force and energy difference are, respectively, less than 0.01 eV/Å and 10-5 eV. Following the previous works,24,36 the spin coupling parameters are calculated based on PBE functional, while HSE06 method is only employed to describe the band structure. In order to more accurately deal with the Coulomb interaction of d electrons, Hey-Scuseria-Ernzerhof hybrid functional (HSE06) are employed.37 Maximally localized Wannier functions (MLWF) are constructed by using the WANNIER90 package.38 Thermal stability are demonstrated with 81 atoms at 300 K based on ab-initio molecule dynamics (AIMD), and the calculations of dynamical stability are carried out with a 3×3×1 supercell based on PHONOPY code.39 3. RESULTS AND DISSCUSSION The crystal structures of SL MP2S6 (M= Cr, Mn, Co) are shown in Figure 1(a). It can be seen that each metal atom is coordinated by six S atoms, forming a distorted octahedral structures. And the P-P pair is perpendicular to the 2D plane. The optimized lattice constants of SL MP2S6 are listed in Table I. In order to estimate their feasibility in experiment, we calculate their binding energies on the basis of the following equation:40-42 Eb= (Etotal-EM-6ES-2EP)/9. 3



Here, Etotal represents the total energy of the system. EM, ES and EP are the chemical potentials of metal atom, S and P atoms, which are calculated from their stable bulk phases. The calculated binding energies for CrP2S6, MnP2S6 and CoP2S6 are, respectively, -1.92, -2.00 and -1.78 eV/atom. These negative binding energies indicate that these compounds might be stable in energy. We also perform the AIMD and phonon spectrum simulations to further evaluate their thermal and dynamical stabilities. As plotted in Figure 1(c) and S1, no imaginary frequencies appear for SL CrP2S6, while for SL CoP2S6 and MnP2S6, little image frequencies occur, which can be deemed to be stable in dynamics. Figure S2 and S3 display the vibration of energy and snapshots of SL MP2S6 after annealing at temperature of 300 K for 3 ps and 500 K for 10 ps. It can be seen that these systems still maintain original structures without bond broken and structural reconstruction. And the variations of total energy for SL MP2S6 are within a small range. Accordingly, the proposed SL MP2S6 is highly expected to be experimentally fabricated. Besides these systems, we also estimate the stabilities of SL FeP2S6 and NiP2S6, and unfortunately, they are not stable.

Figure 1. (a) Top and side views of the crystal structure of SL MP2S6. Insert in (a) is 4


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the first BZ with marking the high symmetry points. (b) The binding energies of SL MP2S6. (c) The phonon dispersion curves of SL CrP2S6. (d) Fluctuations of total energy per unit cell with time obtained from MD simulation of SL CrP2S6 at 300 K. Insert in (d) is the snapshot of SL CrP2S6 at the end of MD simulation at 300 K from side top views. Orange, light blue and gray balls respectively represent S, P and M atoms. Table I. The optimized lattice constants, binding energy (Eb), magnetic moment  per unit cell and the energy difference between FM and AFM states for SL MP2S6. Crystals

CrP2S6

MnP2S6

CoP2S6

Lattice constants (Å)

5.93

5.91

5.77

Eb (eV/atom)

-1.92

-2.00

-1.78

 B

2

3

1

EFM-EAFM-1 (meV)

-256.45

-125.48

-101.13

EFM-EAFM-2 (meV)

-248.43

-107.30

-88.95

As SL MP2S6 contain transition metal elements, it is highly possible to observe the magnetic behaviors in SL MP2S6. To this end, we perform spin-polarized calculations. We find that all SL MP2S6 are magnetic, and the calculated magnetic moment for CrP2S6, MnP2S6 and CoP2S6 are 2, 3 and 1B, respectively. And as shown in Figure S4, the magnetic moments are mainly contributed by the transition metal atoms. Note that the existence of magnetic moment does not guarantee the formation of ferromagnetic (FM) coupling, we further investigate the magnetic ground state for SL MP2S6. We construct a 2 × 2 supercell with four magnetic metal atoms (AFM-1), as shown in Figure 2(b), and considering their triangular lattices, we also investigate the 120° noncollinear antiferromagnetic structures with the magnetic moment directions lying in the xy plane for all these systems. To this end, we construct a

3× 3 supercell with three magnetic metal atoms (AFM-2). As listed in Table I, for all these three systems, the total energies of FM states are obviously smaller than those of their corresponding AFM-2 configurations, thus indicating that SL MP2S6 are characterized 5



with FM ground state. The Curie temperature for magnetic coupling is of extremely significance for the applications in spintronics and for the observation of QAH effect. To estimate the Curie temperature of SL MP2S6, Ising model combining with Monte Carlo (MC) simulation are employed. The spin Hamiltonian can be written as: H= ∑ijJSi∙Sj, where J represents the exchange energy, Si and Sj are respectively the magnetic moments of M atoms at i and j sites. For SL MP2S6, the term J is determined by J = (1/6) Eex/2S2, where Eex = EFM-EAFM, 1/6 denotes the six nearest-neighbor coupling-interaction magnetic atoms, and S is the spin of each metal atom. The calculated J for SL CoP2S6, CrP2S6 and MnP2S6 are 8.43, 5.34 and 1.16 meV, respectively. A 80 × 80 supercell and 106 loops are adopted to carry out the MC simulation. The Curie temperature for SL CoP2S6, CrP2S6 and MnP2S6 are found to be about 200, 350 and 150 K. Particularly for SL CrP2S6, the high Curie temperature makes it meet the requirement of its applications at room temperature. Also, we employ the mean-field theory and the Heisenberg model43,44 kBTc= (2/3)∆E to estimate the Curie temperature and the obtained Tc for SL MP2S6 (M = Co, Cr, Mn) are, respectively, 211K, 535K and 256K, which are a little higher than those based on Ising model.

Figure 2. (a) Variation of magnetic moment per unit cell with temperature for SL MP2S6 based on Monte Carlo simulation. (b) displays the FM and AFM configurations. 6


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In the following, we will focus on the electronic properties of SL MP2S6 based on the estimated FM ground states. The results based on PBE functional are shown in Figure S5, from which we can see when excluding spin-orbit coupling (SOC), SL CoP2S6 and CrP2S6 are metallic with two spin-minority bands crossing the Fermi level at the  point, while SL MnP2S6 is semiconducting. Upon including SOC, a global band gap of 23 meV is opened for SL CrP2S6, while for SL CoP2S6 and MnP2S6, they are still metallic and semiconducting, respectively. In order to accurately describe the band structures of SL MP2S6, HSE06 functional method is employed to treat the Coulomb interaction of the localized d electrons. As shown in Figure 3, the essential electronic properties of these systems have not been altered, except for the increase of the band gaps for SL CrP2S6 and MnP2S6. Notably, the SOC-induced band gap for SL CrP2S6 is 53 meV, which can be survived at room temperature. The large SOC-induced band gaps in SL CrP2S6 can be understood by analyzing the oribital contributions around the Fermi level. As shown in Figure S6, the states around the Fermi level are mainly from px and py orbitals. These orbitals harbor significantly strong SOC strength, which has been well estimated in previously work.45 In addition, for SL CoP2S6, the spin up channel is insulating, while the spin-down channel is conducting. Such a completely separated majority-spin and minority spin bands near the Fermi level are of great importance for 100% spin-injection for spintronic devices. For SL MnP2S6, the moderate band gap of 1.1 eV renders it a natural candidate for future applications in spintronics and optoelectronics.

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Figure 3. Band structures of SL (a) CoP2S6, (b) CrP2S6 and (c) MnP2S6 without SOC and with SOC based on HSE06 level. The dashed black lines represent the Fermi level. Orange and dark blue lines respectively indicate spin-up and spin-down states.

Figure 4. (a) The calculated band edges of semi-infinite sheet of SL CrP2S6 and (b) the integrated anomalous Hall conductance xy in units of e2/h. 8


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Among these three systems, the interesting SOC-induced band evolution in SL CrP2S6 is a typical scenario for the existence of the QAH effect. To confirm the nontrivial topological properties of the SL CrP2S6, we first calculate its Chern number based on Wannier charge centers approach. The maximally localized Wannier functions are established from first-principle calculations with SOC in the WANNIER90 package. The Chern number of SL CrP2S6 can be viewed as the number of electronic charges pumped across unit cell in the course of a cycle,46 which is evaluated from the evolution of the hybrid Wannier charge centers during a timereversal pumping process:47,48 C=

1 𝑒𝑎

[𝑃ℎ𝑒(2𝜋) ― 𝑃ℎ𝑒(0)].

Here, 𝑎 is lattice constant, 𝑃ℎ𝑒(𝑘𝑦) = 𝑒∑𝑛𝑥𝑛(𝑘𝑦) is the electronic polarization, 𝑥𝑛(𝑘𝑦 ) is the Wannier charge centers, which is a smooth function of 𝑘𝑦 for 𝑘𝑦 ∈ [0, 2𝜋]. The calculated Chern number of SL CrP2S6 is nonzero (C = -1), firmly indicating that it is an intrinsic QAH insulator. It is known that the nontrivial QAH state would guarantee the existence of quantized edges states, we then construct a tight-binding (TB) Hamiltonian with the Green’s function method to obtain the edge states using the WANNIERTOOLS package. As shown in Figure 4, one can clearly see the single chiral topologically protected gapless edge state near Fermi level connecting the valence band and conduction band in the bulk band gap. This is consistent with the calculated Chern number. For the feasibility of experimental measurement, we also study its Hall conductivity with the formulate: xy= Ce2/h. The result displays a quantized terrace of -1 e2/h inside the global band gap. And σxy will decrease to a small value when the chemical potential lies out of the band gap. This curve can be observed by Hall resistance measurement at zero magnetic field in experiments. Considering the large nontrivial band gap combined with the high Curie temperature, the SL CrP2S6 would provide a promising platform for exploring the QAH effect at room temperature. And we wish to stress that the temperature for observing QAH effect in SL CrP2S6 is expected to be three orders of magnitude higher than the temperature (

Room Temperature Quantum Anomalous Hall Effect in Single-Layer

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