Adsorption Tuning of Polarity and Magnetism in AgCr2S4 Monolayer

As a recent successfully exfoliated non-van der Waals layered material, AgCrS2 has received a lot of attention. Motivated by its structure-related magnetic and ferroelectric behavior, a theoretical study on its exfoliated monolayer AgCr2S4 has been carried out in the present work. Based on density functional theory, the ground state and magnetic order of monolayer AgCr2S4 have been determined. The centrosymmetry emerges upon two-dimensional confinement and thus eliminates the bulk polarity. Moreover, two-dimensional ferromagnetism appears in the CrS2 layer of AgCr2S4 and can persist up to room temperature. The surface adsorption has also been taken into consideration, which shows a nonmonotonic effect on the ionic conductivity through ion displacement of the interlayer Ag, but has little impact on the layered magnetic structure.

Inspired by graphene, most previous research has focused on 2D layered van der Waals (vdW) materials, whose atomic-thin layers can be easily obtained by mechanical exfoliation due to their weak vdW interlayer bonding [13][14][15]. With the development of 2D research, more feasible approaches emerge. As a supplement to mechanical cleavage, those methods can artificially open a gap between layers of non-vdW materials through selective etching or ionic intercalation. The 2D layer can then be obtained through post-procedure. The typical representatives prepared by chemical etching and intercalation are MXene and AM 2 X 4 , respectively [16,17]. Since then, non-vdW layered materials soon became another emerging branch of 2D materials, especially those with intrinsic ferroelectricity, long-range spin orders, or both. For example, NaCrX 2 with adjustable conductivity and ACr 2 S 4 (A = Li, Na, K, Rb) nanosheets with multiferroic properties have been reported recently [18,19]. AgCrS 2 is one such layered material with both long-range magnetic order and ferroelectricity. This compound was synthesized in 1957 [20]. It is composed of an alternative stacking of edge-sharing octahedra CrS 2 layers and Ag ion layers along the c-axis in a trigonal lattice (R3m) around room temperature. When cooled down to T N (about 40 K), the lattice changes to monoclinic (Cm), accompanied by the emergence of in-plane double stripes (DS) antiferromagnetic (AFM) order [21,22]. The ferroelectricity originates from the off-centering displacement of Ag ions. Several experimental works found that this polarization is closely related to the structural and magnetic transition [21,22]. Recently, the monolayer AgCr 2 S 4 , consisting of a single Ag layer sandwiched between two CrS 2 layers, was successfully exfoliated from AgCrS 2 bulk [17]. Interest has been aroused [23,24], mainly focusing on the magnetic and ferroelectric properties of single-layer AgCr 2 S 4 . However, the specific structure of the monolayer, the possible surface adsorption after peeling, and their effects on material properties have not been well explored.
In this work, based on density functional theory (DFT), the magnetic ground state of bulk AgCrS 2 has been checked. Our calculation results on bulk are consistent with recent experimental observations, which not only ensure the feasibility of our calculation but also form a solid basis for the following study on its monolayer. The structural, magnetic, and electronic properties of AgCr 2 S 4 monolayer have been further studied. Unexpectedly, the polar symmetry inherited from the parent phase could not be preserved during optimization. Moreover, ferromagnetic (FM) order appears in the in-plane Cr triangular lattice with relatively weak interplane AFM coupling. The situation changes when hydrogen adsorption is taken into consideration. After adsorption, the intralayer FM and interlayer AFM ground state remains unchanged, but the structural symmetry is altered along with its ferroelectricity and ionic transport behavior.

Methods
Our DFT calculations were performed using Vienna ab initio Simulation Package (VASP) [25,26]. The electronic interactions were described by projector-augmented-wave (PAW) pseudo-potentials, with semicore states treated as valence states [27]. The exchange and correlation were treated using Perdew-Burke-Ernzerhof (PBE) parametrization of the generalized gradient approximation (GGA) [28]. To properly describe the correlated electrons, the GGA+U method was adopted, and the on-site Hubbard U eff was imposed on Cr's 3d orbitals using the Dudarev approach for all calculations [29]. The plane-wave cutoff energy was set to 500 eV. The Monkhorst−Pack K-point meshes were chosen as 2 × 8 × 4 and 7 × 7 × 1 for bulk and monolayer calculations, respectively. Exchange coefficients and magnetic ground states for the monolayer were estimated based on a 2 × 4 × 1 supercell with various magnetic orders. The convergent criterion for the energy was set to 10 −5 eV, and that of the Hellman-Feynman forces during structural relaxation was 0.01 eV/Å.
In the study of the monolayer structure, a vacuum layer of 20 Å was added along the c-axis direction to avoid the interaction between two neighboring slices. The possible switching paths between different structure phases were evaluated by the nudged elastic band (NEB) method [30]. To estimate the Curie temperature and the temperature evolution of magnetic properties, the Markov-chain Monte Carlo (MC) method with Metropolis algorithm was employed to simulate the magnetic ordering [31]. The MC simulation was performed on a 40 × 40 lattice with periodic boundary conditions, and larger lattices were also tested to confirm the physical results. The simulations were performed with 20,000 equilibration steps and 80,000 averaging steps. All MC simulations are gradually cooled down from the initial disordered state at high temperature to the low temperature under investigation.

AgCrS 2 Bulk Properties
As mentioned above, the low-temperature bulk crystal belongs to the Cm space group without spatial inversion symmetry. Its intrinsic layered feature is illustrated in Figure 1a. The Cr 3+ ions located at the center of CrS 6 octahedra form a triangular magnetic lattice within each CrS 2 layer. Recently, an in-plane collinear magnetic structure has been reported, showing DS pattern (as shown in Figure 1c) with AFM coupling in between [22,32]. The magnetic ordering and structural transition occur simultaneously, accompanied by the emergence of ferroelectricity, indicating the strong connection between magnetic, ferroelectric, and structural properties [32][33][34][35].
To determine the magnetic ground state, three common collinear configurations on the triangular lattice, as depicted in Figure 1c, have been taken into account in addition to the reported DS order. Besides, in order to study the interlayer magnetic coupling, the intralayer FM and interlayer AFM (A-AFM) configuration has also been considered. Our calculation results show that the DS-AFM pattern is indeed the magnetic ground state when U eff is less than 0.8 eV (Figure 1d). The total energy of AFM zigzag and stripe configurations are always higher than that of DS-AFM and are almost insensitive to the U eff value. In contrast, the energy of A-AFM and FM decrease with increasing U eff , showing a similar trend. The A-AFM order is energetically more favorable than FM, and will even replace DS-AFM as the ground state when U eff is larger than 0.8 eV. Obviously, a specific U eff value (i.e., 0.6 eV) is vital in precisely reproducing AgCrS 2 bulk properties. The local magnetic moment increases with the increase in U eff , reaching 2.86 µ B /Cr at 0.6 eV, which is consistent with previously reported value [22]. Moreover, the optimized lattice constants (a = 13.83 Å, b = 3.54 Å, and c = 7.13 Å) are in good agreement with the experimental data [22,24]. Therefore, it will be adopted in the following calculations by default. To determine the magnetic ground state, three common collinear configurations on the triangular lattice, as depicted in Figure 1c, have been taken into account in addition to the reported DS order. Besides, in order to study the interlayer magnetic coupling, the intralayer FM and interlayer AFM (A-AFM) configuration has also been considered. Our calculation results show that the DS-AFM pattern is indeed the magnetic ground state when Ueff is less than 0.8 eV (Figure 1d). The total energy of AFM zigzag and stripe configurations are always higher than that of DS-AFM and are almost insensitive to the Ueff value. In contrast, the energy of A-AFM and FM decrease with increasing Ueff, showing a similar trend. The A-AFM order is energetically more favorable than FM, and will even replace DS-AFM as the ground state when Ueff is larger than 0.8 eV. Obviously, a specific Ueff value (i.e., 0.6 eV) is vital in precisely reproducing AgCrS2 bulk properties. The local magnetic moment increases with the increase in Ueff, reaching 2.86 µB/Cr at 0.6 eV, which is consistent with previously reported value [22]. Moreover, the optimized lattice constants (a = 13.83 Å, b = 3.54 Å, and c = 7.13 Å) are in good agreement with the experimental data [22,24]. Therefore, it will be adopted in the following calculations by default.

AgCr2S4 Monolayer
Recently, AgCrS2 was successfully exfoliated into 2D nanosheets by Peng et al. through ion intercalation [17]. These 2D sheets can be thinned down to a monolayer containing one Ag ion layer sandwiched between two CrS2 layers. As can be seen from Figure  2a,b, the edge-sharing octahedral framework is inherited in CrS2 layers, while the relative displacement of the center Ag ion will give rise to two distinct structural phases (i.e., the asymmetric α phase and the centrosymmetric β phase). The detailed structural information is shown in Table S1 and Figure S1. In the following, we will focus on the structural, magnetic, and electronic properties of AgCr2S4 monolayer.

AgCr 2 S 4 Monolayer
Recently, AgCrS 2 was successfully exfoliated into 2D nanosheets by Peng et al. through ion intercalation [17]. These 2D sheets can be thinned down to a monolayer containing one Ag ion layer sandwiched between two CrS 2 layers. As can be seen from Figure 2a,b, the edge-sharing octahedral framework is inherited in CrS 2 layers, while the relative displacement of the center Ag ion will give rise to two distinct structural phases (i.e., the asymmetric α phase and the centrosymmetric β phase). The detailed structural information is shown in Table S1 and Figure S1. In the following, we will focus on the structural, magnetic, and electronic properties of AgCr 2 S 4 monolayer.
To find the ground state of AgCr 2 S 4 monolayer, the total energies of different magnetic orders mentioned earlier have been calculated based on the above two structural phases. The calculation results have been summarized in Table 1. Obviously, the energy of the β phase is always lower than that of the α phase, presenting a clear tendency to restore the central symmetry of the 2D single layer, contrary to its parent bulk phase. Table 1. The energy differences of four in-plane collinear spin configurations. Interlayer coupling has also been considered, in which the AFM interlayer coupling is denoted by subscript 1 and the FM coupling is represented by subscript 2, respectively. The total energy of the ground state (β phase with A-AFM order) is taken as the reference value, in units of meV/Cr.  To find the ground state of AgCr2S4 monolayer, the total energies of different magnetic orders mentioned earlier have been calculated based on the above two structural phases. The calculation results have been summarized in Table 1. Obviously, the energy of the β phase is always lower than that of the α phase, presenting a clear tendency to restore the central symmetry of the 2D single layer, contrary to its parent bulk phase. Table 1. The energy differences of four in-plane collinear spin configurations. Interlayer coupling has also been considered, in which the AFM interlayer coupling is denoted by subscript 1 and the FM coupling is represented by subscript 2, respectively. The total energy of the ground state (β phase with A-AFM order) is taken as the reference value, in units of meV/Cr. Our results also indicate that the magnetic Cr ions in each CrS2 layer tend to couple ferromagnetically and show no sensitivity to the above two structural phases. This structure-insensitive FM behavior, in contrast with its bulk form, can be reasonably interpreted on the basis of the d orbital occupation of Cr. In the AgCrS2 bulk, Cr 3+ ion is in the 3d 3 configuration. According to the Goodenough-Kanamori-Anderson (GKA) rules [36][37][38], the half-filled t2g orbitals give rise to AFM direct exchange, while the p orbital intermediated Cr-S-Cr super exchange favors FM coupling. It is the competing exchange interactions that make the bulk magnetic order structurally related. In contrast, from the change of chemical formula before and after exfoliation, the Cr ion in AgCr2S4 monolayer is in the mixed valence state (Cr2 7+ ). The hole hopping between neighboring Cr's d orbitals gives rise to the strong FM tendency and is insensitive to structural details.

A-AFM
To characterize this in-plane triangular magnetic lattice, the classical Heisenberg spin model is adopted, which can be constructed as where Si is the normalized spin (|S| = 1) on the Cr site i. J1 and J2 correspond to the inplane exchange constants between the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) interactions, as labeled in Figure 2b, respectively. Kb/c stands for the single-ion Our results also indicate that the magnetic Cr ions in each CrS 2 layer tend to couple ferromagnetically and show no sensitivity to the above two structural phases. This structureinsensitive FM behavior, in contrast with its bulk form, can be reasonably interpreted on the basis of the d orbital occupation of Cr. In the AgCrS 2 bulk, Cr 3+ ion is in the 3d 3 configuration. According to the Goodenough-Kanamori-Anderson (GKA) rules [36][37][38], the half-filled t 2g orbitals give rise to AFM direct exchange, while the p orbital intermediated Cr-S-Cr super exchange favors FM coupling. It is the competing exchange interactions that make the bulk magnetic order structurally related. In contrast, from the change of chemical formula before and after exfoliation, the Cr ion in AgCr 2 S 4 monolayer is in the mixed valence state (Cr 2 7+ ). The hole hopping between neighboring Cr's d orbitals gives rise to the strong FM tendency and is insensitive to structural details.
To characterize this in-plane triangular magnetic lattice, the classical Heisenberg spin model is adopted, which can be constructed as where S i is the normalized spin (|S| = 1) on the Cr site i. J 1 and J 2 correspond to the in-plane exchange constants between the nearest-neighbor (NN) and the next-nearestneighbor (NNN) interactions, as labeled in Figure 2b, respectively. K b/c stands for the single-ion magnetocrystalline anisotropy along the b-/c-axis, respectively. Based on the ground structure (β phase), these exchange coefficients can be extracted by comparing DFT energy with different spin orders. Specifically, in a 2 × 4 × 1 supercell, the energy of these magnetic states can be expressed as where E 0 is the nonmagnetic energy. The derived parameters are summarized in Table 2. According to our estimation, the NN exchange interactions are FM and dominated, as expected from our previous analysis. The NNN exchange constant J 2 is relatively weak due to its indirect and long-distance bonding. Based on these exchange parameters, MC simulations were employed to determine the Curie temperature. Additionally, the magnetic susceptibility was calculated. The system reaches equilibrium at a given temperature, the magnetization M and magnetic susceptibility χ are calculated as [39] where N represents the total number of spin sites. Given the exact solution of the spin Hamiltonian, T C can be estimated from the peak position of the specific magnetic susceptibility χ (or the maximum slop point of magnetization M). MC results are shown in Figure 2c, indicating that the magnetic transition temperature is above room temperature, much higher than its parent bulk, as expected from its changed and mixed valence of Cr ions. Our MC simulation have also been tested on multi-size lattices to exclude the finite size effect. As shown in Figure S2, the magnetization and susceptibility curve are not sensitive to lattice size, and the Curie temperature shows no obvious scale effect. Since the AgCr 2 S 4 monolayer contains two CrS 2 layers, the interlayer coupling has also been considered. Our calculation shows that the interlayer AFM coupling is energetically more favorable than FM coupling, although the energy difference is quite limited (within 3 meV). After exfoliation, the CrS 2 -CrS 2 interlayer coupling decreases with the increase in the interlayer spacing (from 4.60 Å to 4.76 Å), which is consistent with the intuition. These weakly coupled 2D FM triangular lattices in the AgCr 2 S 4 monolayer may provide a new approach for magnetic regulation in spintronic devices.
The electronic densities of states (DOS) of AgCrS 2 bulk and AgCr 2 S 4 monolayer are shown in Figure 3. The parent bulk phase exhibits insulating characteristics with a moderate gap of about 1.5 eV. The states near the Fermi level mainly originate from Cr's 3d orbitals. In the AgCr 2 S 4 monolayer, the stripping induced hole-doping causes a negative shift in the Fermi level, and therefore closes the gap. Meanwhile, Cr's dominant contribution to the Fermi level is not affected upon peeling.
In addition, we also verified the mechanical stability of AgCr 2 S 4 . The in-plane elastic constants and various mechanical parameters are summarized in Tables S2 and S3, respectively. Our calculation results prove that AgCr 2 S 4 is a soft and malleable material, similar to its three-dimensional counterpart [40]. The schematic diagrams of Young's modulus, shear modulus, and Poisson's ratio are given in Figure S5.

H Adsorption Effect
The AgCr 2 S 4 monolayer is synthesized through wet chemical exfoliation of bulk AgCrS 2 . Analogous to MXene, the high surface area to volume ratio and the unsaturated bonds of the outer layer sulfur ions may lead to the adsorption of ions at surface sites during preparation. Thus, the hydrogen adsorption and its effect on structural and magnetic properties of AgCr 2 S 4 monolayer have been investigated. First, the unilateral passivation is considered. As labeled in Figure 4a,b, there are ten possible adsorption sites. According to our calculations, the energetically most favorable adsorption site (denoted as C in Figure 4a) is located right above the sulfur anion. Detailed adsorption sites and adsorption energy are provided in the supplementary material. After unilateral hydrogen passivation, the central Ag layer shifts away from the adsorption side due to electrostatic repulsion, recovering the original bulk-like AgS 4 tetrahedron with neighboring S ions.

H Adsorption Effect
The AgCr2S4 monolayer is synthesized through wet chemical exfoliation of bulk AgCrS2. Analogous to MXene, the high surface area to volume ratio and the unsaturated bonds of the outer layer sulfur ions may lead to the adsorption of ions at surface sites during preparation. Thus, the hydrogen adsorption and its effect on structural and magnetic properties of AgCr2S4 monolayer have been investigated. First, the unilateral passivation is considered. As labeled in Figure 4a,b, there are ten possible adsorption sites. According to our calculations, the energetically most favorable adsorption site (denoted as C in Figure 4a) is located right above the sulfur anion. Detailed adsorption sites and adsorption energy are provided in the supplementary material. After unilateral hydrogen passivation, the central Ag layer shifts away from the adsorption side due to electrostatic repulsion, recovering the original bulk-like AgS4 tetrahedron with neighboring S ions.

H Adsorption Effect
The AgCr2S4 monolayer is synthesized through wet chemical exfoliation of bulk AgCrS2. Analogous to MXene, the high surface area to volume ratio and the unsaturated bonds of the outer layer sulfur ions may lead to the adsorption of ions at surface sites during preparation. Thus, the hydrogen adsorption and its effect on structural and magnetic properties of AgCr2S4 monolayer have been investigated. First, the unilateral passivation is considered. As labeled in Figure 4a,b, there are ten possible adsorption sites. According to our calculations, the energetically most favorable adsorption site (denoted as C in Figure 4a) is located right above the sulfur anion. Detailed adsorption sites and adsorption energy are provided in the supplementary material. After unilateral hydrogen passivation, the central Ag layer shifts away from the adsorption side due to electrostatic repulsion, recovering the original bulk-like AgS4 tetrahedron with neighboring S ions.  Nonetheless, the magnetic ground state of AgCr 2 S 4 H remains A-AFM (e.g., in-plane FM). It is nontrivial, since the chemical valence of Cr has been restored to its bulk value (i.e., +3) after the hydrogen adsorption. The in-plane DS-AFM order observed in bulk has not been recovered, which is a little bit unexpected. As discussed previously, the inplane magnetic pattern in bulk is structurally related, namely the Cr 3+ ion spacing [22,35]. We compared the in-plane Cr-Cr spacing after unilateral H passivation with that of the bulk material. The Cr-Cr spacing after passivation is 3.55 Å larger than the bulk value (3.43/3.54 Å, this non-uniform spacing distribution is due to the DS-AFM order). According to GKA rules, the half-filled t 2g orbitals of Cr 3+ give rise to the AFM direct exchange which is sensitive to Cr-Cr spacing. In other words, large spacing weakens this AFM direct exchange, breaks the delicate balance and leads to the FM dominance, which well explains the FM behavior observed in the passivated AgCr 2 S 4 H material. Details of lattice constants are marked in Figure S4. This speculation is further proved by in-plane strain modulation. As shown in Figure 4c, the in-plane magnetic ground state is fragile and extremely sensitive to biaxial strain. The compressive strains can effectively shorten the in-plane Cr-Cr distance and enhance the direct AFM coupling, thus giving rise to the bulk-like DS order. On the contrary, the tensile strains will fasten the in-plane FM order (i.e., A-AFM).
In AgCrS 2 bulk, the sandwiched Ag layer has been proved to be crucial to its structure, ferroelectricity, and ionic conductivity [17,20,33]. Here, in AgCr 2 S 4 monolayer, Ag's displacement to the central site restores the centrosymmetry and destroys the ferroelectricity. This displacement not only increases the CrS 2 interlayer distance, but also weakens the binding between Ag ion and the upper/lower CrS 2 layers (Ag-S bonds are halved from 4 to 2), which will certainly lead to the enhancement of ionic mobility as found in experiment [20].
To confirm this scenario, the possible displacement processes of Ag ion in AgCr 2 S 4 as well as its H-passivated cases are simulated by the NEB method. The corresponding energy profiles are shown in Figure 5. Obviously, unilateral adsorption breaks the centro-symmetry and forces Ag to shift away from H, resulting in an asymmetric potential profile, which is consistent with our previous analysis. The simulation results of the AgCr 2 S 4 monolayer and the bilaterally adsorbed AgCr 2 S 4 H 2 are also presented in Figure 5 for comparison. In both AgCr 2 S 2 and AgCr 2 S 4 H 2 , the sandwiched Ag ion tends to be located in the central site, with weak bonding between neighboring S ions. The energy barriers of AgCr 2 S 2 and AgCr 2 S 4 H 2 are 150 meV/Ag and 140 meV/Ag, respectively, making them much lower than in the case of AgCr 2 S 4 H (190 meV/Ag). For comparison, the energy barrier of AgCrS 2 bulk was estimated to be 450 meV/Ag [17].

Conclusions
In summary, the structural, magnetic, and electronic properties of bulk AgCrS2 and its single layer AgCr2S4 have been investigated. The in-plane DS pattern of bulk material has been confirmed, but its monolayer exhibits metallic in-plane FM behavior. The underlying mechanism is attributed to the hole-doping induced by exfoliation and the resulting change in Cr's chemical valence. Moreover, the displacement of the sandwiched Ag ion towards the high-symmetry center was found, which eliminates the polarity and enhances the ionic conductivity. This structure-related superionic behavior is sensitive to surface adsorption. Specifically, it will be inhibited by unilateral adsorption, but will be recovered by bilateral adsorption. The present study may stimulate further experimental and theoretical research on AgCr2S4 and other 2D non-vdW materials. Based on the above analysis and numerical data, it is reasonable to conclude that the ionic conductivity of AgCrS 2 can benefit from dimension reduction and may reach its peak in single-layer AgCr 2 S 4 or its AgCr 2 S 4 H 2 .

Conclusions
In summary, the structural, magnetic, and electronic properties of bulk AgCrS 2 and its single layer AgCr 2 S 4 have been investigated. The in-plane DS pattern of bulk material has been confirmed, but its monolayer exhibits metallic in-plane FM behavior. The underlying mechanism is attributed to the hole-doping induced by exfoliation and the resulting change in Cr's chemical valence. Moreover, the displacement of the sandwiched Ag ion towards the high-symmetry center was found, which eliminates the polarity and enhances the ionic conductivity. This structure-related superionic behavior is sensitive to surface adsorption. Specifically, it will be inhibited by unilateral adsorption, but will be recovered by bilateral adsorption. The present study may stimulate further experimental and theoretical research on AgCr 2 S 4 and other 2D non-vdW materials.

Informed Consent Statement: Not applicable.
Data Availability Statement: Data supporting these findings are available from the corresponding authors upon request.