Intrinsic two-dimensional multiferroicity in CrNCl2 monolayer*

Wei Shen(沈威) Yuanhui Pan(潘远辉) Shengnan Shen(申胜男)Hui Li(李辉) Siyuan Nie(聂思媛) and Jie Mei(梅杰)

1School of Power and Mechanical Engineering,Wuhan University,Wuhan 430072,China

2Research Institute of Wuhan University in Shenzhen,Shenzhen 518057,China

3Department of Quantum and Energy Materials,International Iberian Nanotechnology Laboratory,Braga 4715-330,Portugal

4School of Logistic Engineering,Wuhan University of Technology,Wuhan 430070,China

Keywords: two-dimensional multiferroic CrNCl2 monolayer magnetism ferroelectricity

1. Introduction

Nanoscale spin-based electronics(or spintronics),[1,2]using the spins of electrons for future generation of information storage, transmission and processing with the promising advantages of high storage density,high integration density,low energy consumption and high device operation speed,[3,4]are drawing more and more research interests these years. Unfortunately,most of those explored two-dimensional(2D)materials,such as graphene,[5]h-BN[6]and phosphorene,[7]possess none or weak intrinsic ferromagnetic (FM) ordering, which block their direct applications in spintronics. It is reported that magnetism in 2D materials can be induced by the process of adsorption,[8]doping,[9]defect engineering[10]and chemical functionalization,[11]the artificially induced magnetism,however, until now it has been observed to be rather weak.[12,13]For 2D ferromagnetic materials used in spintronic devices,the coexistence of a large spin polarization and a high enough Curie temperature is of great interest and importance.[4,14,15]So far, a few 2D materials have been reported to be potential in spintronic devices, such as Cr2Ge2Te6.[14]MnN,[16]CoH2,[17]and Cr3Te4,[18]and there are more 2D materials to be discovered. Analogous to ferromagnetism, ferroelectric(FE)materials are those with spontaneous electric polarization which is switchable by an external electric field.[19]FE materials have promising potentials in applications of non-volatile memories,[20,21]fieldeffect transistors,[22,23]sensors and photonics devices.[24-26]Meanwhile,ultrathin FE films are clearly needed in today’s industrial applications, in a necessary attempt to improve the integrity and energy efficiency.[27-30]However, when a three-dimensional FE perovskite oxide is thinned down to a critical thickness of around several to tens of nanometers, its spontaneous polarization usually disappears,because of the reduced long-range Coulomb coupling, depolarizing electrostatic field, the effects of surface energy and electron screening.[27,29,30]Up to date, a few 2D materials have been revealed or predicted to exhibit stable ferroelectricity, including 1T-MoS2,[31]SnTe[32]and CuInP2S6.[33]Still,plenty of 2D ferroelectrics are out there waiting to be explored.

Multiferroics,which simultaneously exhibit ferroelectricity and magnetism in a single phase, have been intriguing intensive interests due to their novel physics and promising applications in memories and spintronics.[34-36]There are two kinds of multiferroics that have been reported. In type-I multiferroics,the FE and FM orders originate from different sources and meanwhile they usually couple weakly. However,the ferroelectricity is resulted from the FM order which breaks the centrosymmetry in type-II multiferroics. In type-II multiferroics, the magnetoelectric coupling is usually strong, and yet their FE orders are often very weak and the Curie temperature is rather low.[37,38]Therefore, to reach ideal multiferroics, one direction is to merge their strength together in one material. Although with many promising and appealing applications, only a few 2D multiferroic candidates have been reported so far, such as magnetic-FE Hf2VC2F2,[39]FM-FE electron-doped CrBr3monolayer,[40]VOCl2monolayer[41,42]and Fe2O3monolayer.[43]However,scientific evidences of intrinsic multiferroicity in 2D materials without artificial treatments like surface treatment and doping are rather rare so far.In this work, based upon the first-principles calculations, we propose an intrinsic multiferroic,CrNCl2monolayer.

2. Method

All of the calculations are carried out based on the density functional theory (DFT) as implemented in the Viennaab initiosimulation package (VASP)[44,45]within the projector augmented-wave method.[46]General gradient approximation in the Perdew-Burke-Ernzerhof[47]expression is chosen as the exchangecorrelation functional. The plane-wave cutoff energy is set to be 600 eV.A vacuum space of 15 ˚A is added between each monolayer to avoid the spurious interaction between each layer and its periodic replicate. Thek-points grids of 12×12×1 and 6×6×1 under the Monkhorst-Packkpoints scheme[48]are employed to sample the Brillouin zone for the unit cell and 2×2×1 supercell,respectively. The convergence criteria are set to be 0.005 eV/˚A for the force during the structural relaxation and 10−6eV for energy. The convergence of the setting has been tested and verified. Firstprinciples molecular dynamics (MD) simulations are carried out using the canonical (NVT) ensemble[49]with a time step of 1.0 fs for 10 ps. The polarization calculation is performed by the Berry phase method,[50]in which both the electronic and ionic contributions are taken into account. The minimum energy paths of FE transitions are deduced by the climbing image nudged elastic band(CINEB)method,[51]which is based upon the interatomic forces and total energies calculated from the DFT simulations. The Dudarev implementation[52]is used with an on-site Coulomb interactionUeff=U −Japplied to the 3d orbitals. A detailed calculation on differentUeffis listed in Table S2 in the supporting information.

3. Results and discussion

3.1. Geometry and stability

Firstly, the energetic ground-state of CrNCl2monolayer is searched. Inspired by the newly reported VOCl2,[41,42]the CrNCl2monolayer is designed with a similar structure,so are CrNBr2and CrNI2. Through geometry optimization without any symmetry constraints,the stable CrNCl2monolayer structure is found,and computational details are shown in the supporting information. In the following study, the electronic,magnetic and ferroelectric properties of the CrNCl2monolayer will be focused.

The optimized atomic structure of CrNCl2monolayer is illustrated in Fig. 1. The black dashed rectangle in Fig. 1(a)denotes the structural unit cell, and it exhibitsPmm2 symmetry. As depicted in Fig. 1(b), each Cr atom connects to two N atoms and four halogen atoms, showing an octahedral framework. Like VOCl2proposed by Aiet al.,[41]there is no central inversion symmetry in the CrNCl2monolayer. The Cr atom moves towards one of the bridging N atoms along the basis vectora(as thez-direction), and away from the octahedral center by about 0.42 ˚A. The two Cr-N bonds are thus not equivalent in the CrNCl2monolayer.The optimized lattice constants and bond lengths of CrNX2(X=Cl, Br, and I) are listed in Table S1.

Fig.1. Crystal structures of the CrNCl2 monolayer: (a)top view along the c axis,side view along the b axis and(b)along the a axis of the supercell.The dashed rectangle indicates the primitive cell. (c)The phonon spectra of the CrNCl2 monolayer. (d)The MD simulations of the CrNCl2 monolayer at a temperature of 500 K for 10 ps. The inset show the side view of the CrNCl2 monolayer at the end of the simulations.

Then the dynamical behavior and thermal stability of the CrNCl2monolayer are studied. From the phonon spectra in Fig. 1(c), it is found that there are no imaginary-frequency modes, and therefore the CrNCl2monolayer is dynamically stable.In addition,to verify the thermal stability of the CrNCl2monolayer, the MD simulations are performed at a temperature of 500 K for 10 ps with a time step of 1.0 fs. As shown in Fig.1(d),the structure of the CrNCl2monolayer is presented in this timescale,indicating the thermodynamic stability of the CrNCl2monolayer at a temperature of up to 500 K.The structural stability of CrNBr2and CrNI2monolayers is verified in the supporting information(Figs.S1 and S2).

3.2. Magnetic properties

Here four magnetic configurations (AFM1, AMF2,AFM3 and FM) are considered, as depicted in Figs. 2(a) and 2(b)to find the magnetic ground state(MGS).The results are listed in Table 1. The MGSs of CrNCl2, CrNBr2, and CrNI2monolayers are AFM1,AFM1 and FM,respectively. The unit cell of the CrNCl2monolayer is found to be spin-polarized with a total magnetic moment of～1.0µBmainly coming from the dxyorbital of the Cr5+cation,as shown in Fig.2(c).

Fig.2. (a)Top and side views of the spin density isosurfaces of CrNCl2 monolayer with AFM1 order. The solid and dashed arrows refer to the direct d-d exchange and halogen-mediated super exchange,respectively,between the moments on adjacent Cr. (b)Top views of the spin density isosurfaces of CrNCl2 monolayer with AMF2, AFM3 and FM orders.Yellow and blue isosurfaces represent positive and negative spin density, respectively, at the absolute value of 0.3 e/˚A3. (c) Projected density of states (PDOS) of the unit cell in CrNCl2 monolayer with AFM1 order. The valence band maximum is set as zero here.

Table 1. Calculated total energies E (meV) relative to the energy of AFM1 configuration (EAFM1) of four magnetic orders in 2×2 supercells of CrNCl2,CrNBr2 and CrNI2 monolayers.

In fact, these three monolayers are very similar aside from the radii of different halogen atoms. To further explain their magnetic orders,the magnetic exchange paths along Cr-Cl bond chains (alike for CrNBr2and CrNI2) are shown in Fig. 2(a). There are two kinds of exchange interactions here.One is direct d-d exchange(shown by solid arrows)between the local moments of adjacent Cr atoms, which will lead to antiparallel spin alignment. The other is the halogen-mediated super exchange(shown by dashed arrows),and it favors a parallel spin alignment according to the Goodenough-Kanamori rule.[43,53]The dominant one of these two interactions will determine the overall magnetic order of the 2D system. It is noteworthy that in systems with larger halogen radius, the physical separation between Cr atoms alongb-direction will also be larger,which will weaken the direct exchange. Table 1 shows that for CrNCl2and CrNBr2,the direct exchange interaction still dominates the magnetic orders at relatively smaller halogen radii, and leads to AFM orders. While in CrNI2, the system shows an FM order with the direct exchange decreasing to be less powerful than the super exchange at such a radius of the halogen I. The weaker direct exchange in CrNI2is due to the largerb-direction distance than that of CrNCl2or CrNBr2(see Table S1).The larger halogen-mediated super exchange in CrNI2can be explained by the fact that an increased p-d (p orbitals from halogen; d orbitals from Cr) hybridization than that of CrNCl2or CrNBr2,as shown in Fig.2(c)and Fig.S4(b). A test onUeffcan be found in Table S2.

Monte Carlo simulations on the basis of the 2D Ising Hamiltonian model have been performed to examine the spin dynamics (computational details are shown in the supporting information). The CrNCl2monolayer has an estimated N´eel temperatureTNof 88 K, which is higher than the Curie temperature of 2D Cr2Ge2Te6(20 K),[14]2D CrI3(45 K)[15]and the boiling point of liquid nitrogen(77 K).However,the N´eel temperature of CrNCl2is lower than VOCl2.[41]The N´eel temperature of CrNBr2monolayer is estimated to be 56 K,which is lower than CrNCl2. The Curie temperature of CrNI2monolayer is estimated to be 38 K. Therefore, the CrNCl2is more suitable for non-volatile memory devices among the CrNX2because of its higher N´eel temperature.

3.3. Ferroelectric properties

We turn to investigate the intrinsic ferroelectricity of the CrNCl2monolayer. Usually, a sizable spontaneous polarization is resulted from the breaking of the inversion symmetry in the CrNCl2monolayer because of the off-centering displacement of the Cr ion in the octahedron.In the attempt to evaluate the spontaneous polarization,we choose an adiabatic pathway from FE phase, through paraelectric (PE) phase, and to the FE phase with an opposite polarization direction(as the inset of Fig. 3(b)) by using the Berry phase method. The calculations reveal a spontaneous in-plane polarization along theadirection, as shown in Fig. 4(a). In addition, the double well potential of the CrNCl2monolayer is also depicted in Fig.4(b)to show the spontaneous polarization along the adiabatic path.The CrNCl2monolayer is calculated to exhibit a spontaneous polarization of 200 pC/m,which is in the same order of magnitude with those reports of other 2D ferroelectrics(e.g.,monolayer group-IV monochalcogenides[28]) and 2D multiferroics(e.g., VOCl2[41]). If the thickness of CrNCl2monolayer is treated as 1 nm, then the effective polarization in bulk state will be equal to 20.0µC/cm2,which is also in the same order of magnitude as those traditional multiferroics materials such as BiFeO3thin film heterostructures.[54]Therefore, CrNCl2can be a candidate in logic and memory devices.Also,the calculated spontaneous polarizations of CrNBr2and CrNI2are 339 and 310 pC/m,respectively. Furthermore,the total energy of CrNCl2as a function of spontaneous polarization exhibits a characteristic double-well potential of ferroelectrics,as displayed in Fig.4(b). The value of the ferroelectric double-well barrier is 1.45 eV/unit-cell,and it is too large for the ferroelectric polarization switching.Therefore,the uniaxial strain is applied on monolayer CrNCl2in order to reduce the double-well barrier. The calculation results are presented in the supporting information.

Fig.3. The specific heat capacity versus temperature for CrNCl2 monolayers, obtained from Monte Carlo simulations on the basis of the 2D Ising Hamiltonian model. N´eel temperature is estimated to be 88 K.

The activation barrier reversing in-plane polarization is shown in Fig. 4(c). An intermediate antiferroelectric (AFE)phase is in the middle of the pathway. The energy of AFE phase is 11.3 meV/unit-cell larger than that of FE phase, and the energy barrier between these two phases is as large as 0.68 eV/unit-cell, which makes the FE phase to be robust against thermal fluctuation. This energy barrier of CrNCl2is much larger than that (0.26 eV) of ferroelectric LiNbO3[55]and that(0.18 eV)of VOCl2,[41]and it indicates that CrNCl2needs much higher electric field to switch the polarization.

Fig.4. (a)Total polarization as a function of normalized displacement where the centrosymmetric paraelectric phase(0%displacement)is at the center,and two ferroelectric ground states are at two ends. (b)Doublewell potential of monolayer CrNCl2 along the switching path. (c)Kinetics pathways of the polarization reversal processes in the CrNCl2 monolayer calculated by the CINEB method. The transformation proceeds through a centrosymmetric structure in which the polarization is zero.

The d rule in multiferroic says that generally, the transition metal d electrons have an important role in the occurrence of magnetism.[56]However, those d electrons are believed to suppress the off-center ferroelectric distortion.[56]Then, why would the proposed CrNCl2, CrNBr2and CrNI2, whose FE characteristics come directly from the Cr5+cation which has a dxyelectron,exhibit FM and FE properties at the same time,violating the conventional d0rule? As shown in Figs. 2(a)and 2(b),the plane of the dxyorbital of the Cr5+cation is perpendicular to the direction of the FE polarization, which, as a result, hardly suppresses the occurrence of ferroelectricity,and the detailed analysis can be found in the supporting information. This is the similar case with VOX2(X=Cl, Br and I).Therefore,the proposed CrNCl2,CrNBr2and CrNI2monolayers are calculated to combine the advantages of type-I and type-II multiferroics, showing large polarizations and magnetoelectric coupling. The calculations enrich the search of multiferroics and inspire further research beyond the d rule.

4. Conclusions

In summary, based on first-principles calculations, we have proposed an intrinsic multiferroic, i.e., a CrNCl2monolayer. It has an antiferromagnetic ground state,with N´eel temperature of about 88 K,and it exhibits an in-plane spontaneous polarization of 200 pC/m. The energy barrier reversing polarization is about 0.68 eV/unit-cell, which makes the FE phase to be robust against thermal fluctuation. The dxyorbital of the Cr atom is the origin of magnetic moments but dxyorbital’s plane is perpendicular to the direction of the FE polarization, which hardly suppresses the occurrence of ferroelectricity.In addition,like CrNCl2,the CrNBr2is also an intrinsic multiferroic with antiferromagnetic-ferroelectric ground state while CrNI2is an intrinsic multiferroic with ferromagneticferroelectric ground states. With the promising properties shown above and in the main text, the CrNX2multiferroic monolayers show a great potential in applications of nanoscale data storage and spintronics. It is also worth noting that further experimental research is welcomed to build up full and detailed understanding of these 2D multiferroics.