A strong coupling mechanism between ferromagnetism and piezoelectricity in 2D ferroelectric CrXSYBrZ with high carrier mobility†

Xiao Shang^ab, Gui-Juan Du^a, Jun-Hui Wang^b, Dan-Yang Zhu^a, Fu-Chun Liu*^a, Xi-Zhe Liu^a, Zeng-Tao Lv^b, Feng Guo^b and Xiao-Chun Wang*^b
aInstitute of Atomic and Molecular Physics, Jilin university, Changchun 130012, China. E-mail: lfc@jlu.edu.cn
^bSchool of Physics Science and Information Technology, Liaocheng University, Liaocheng 252000, China. E-mail: wangxiaochun@tsinghua.org.cn

First published on 14th August 2024

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1. Introduction

Electromechanical coupling is essential for sensing,^1,2 actuating,^3,4 and energy harvesting applications.^5,6 Piezoelectric materials, which can be electrically polarized by external strain and deformed by an applied voltage,^7,8 play a crucial role in converting signals between mechanical and electrical domains. This unique property of piezoelectricity facilitates seamless information transfer between mechanical and electrical systems, improving the efficiency and functionality of various devices. Piezoelectricity can only occur in materials with non-centrosymmetric structures and a bandgap, meaning that they are non-metallic.⁹ Two-dimensional (2D) ultrathin semiconductors, such as monolayer α-In₂Se₃ with a large out-of-plane piezoelectric coefficient d₃₃ = 0.34 pm V⁻¹, which have been widely used for sensors, actuators, electronics, and energy conversion.¹⁰ 2D piezoelectric materials are commonly derived from bulk materials through exfoliation. The central symmetry of the monolayer is more likely broken in comparison to their bulk counterparts, which often results in excellent piezoelectricity exhibited by many 2D materials. This phenomenon is commonly observed in systems such as hexagonal boron nitride (h-BN), transition-metal dichalcogenides (TMDCs) and dioxides (TMDOs), II–VI semiconductors (SCs), III–V SCs, III-monochalcogenides (III-MCs) and IV-monochalcogenides (IV-MCs).^11,12 The advantage of 2D piezoelectric materials lies in their expansive in-plane area. When a significant accumulation of polarized charges occurs on their surfaces, the application value of 2D piezoelectric materials becomes remarkably significant. This requirement highlights the necessity for these materials to exhibit excellent out-of-plane piezoelectricity. Unfortunately, the majority of 2D piezoelectric materials demonstrate solely in-plane piezoelectricity, attributed to their complete symmetry in the vertical direction (perpendicular to the material plane). This limitation stands as a significant obstacle impeding the widespread utilization of 2D piezoelectric materials.^13,14 Recently, 2D Janus magnetic materials like antiferromagnetic Fe₂BrMgP and Cr₂CH₂ have been shown to exhibit the valley Hall effect and are thus of considerable interest for both fundamental condensed matter physics and device engineering.^15,16 2D materials with various special properties are defined as 2D multifunctional materials. Among them, 2D piezoelectric ferromagnetic materials have attracted wide attention like 2D Janus FeGeN₃, which exhibits remarkable out-of-plane piezoelectricity (d₃₁ = −0.31 pm V⁻¹) and robust ferromagnetic order with a Curie temperature of 302 K at the same time.¹⁷ These intriguing properties make Janus FeGeN₃ an attractive candidate for spintronic devices with multifunctionality. Furthermore, the electric potential difference antiferromagnetism in 2D Janus Mn₂ClF can even be tuned by the piezoelectric effect,¹⁸ which manifests that magnetism and piezoelectricity interact in 2D materials.

The current research has achieved the successful exfoliation of the CrSBr monolayer from its bulk form. Both experimental and theoretical investigations have verified that CrSBr acts as a semiconductor with distinctive physical characteristics, including ferromagnetism, tunable dielectric behavior, and spin–phonon coupling.^19–21 Besides, the CrSBr monolayer shows half-metallic properties within an appropriate electric field range. The Cr₂S₂BrI monolayer obtained by atom replacement based on the CrSBr monolayer has an internal electric field due to the different electronegativity of the Br atom and I atom. When the applied electric field value is less than its internal electric field, the Cr₂S₂BrI monolayer retains the semiconductor property and its band gap can be regulated by the applied electric field.²² Regrettably, CrSBr lacks piezoelectricity due to its complete symmetry. To expand its versatility, we have introduced atomic displacement within CrSBr to disrupt its vertical symmetry.^23–26 This approach leads to the creation of five 2D semiconductors, which are named CrXSYBrZ (where X = Cr or Mo, Y = S or Se and Z = H, F, Cl or Br), showcasing ferromagnetism, excellent out-of-plane piezoelectricity and high carrier mobility. These 2D multifunctional materials hold promise for applications in nanodevices, spintronics and photovoltaic cells by applying the strain to engineer the magnetic moment.^27–29 Furthermore, we have introduced an innovative physical mechanism for evaluating the extent of vertical asymmetry charge transfer in CrSBr and CrXSYBrZ (six atomic layers monolayers), which is extended from the electronegativity difference ratio that only adapts to 2D materials consisting of three atom layers and defined as the electronegativity difference effect. This effect allows us to delve deeper into the factors contributing to the vertical mirror symmetry of the CrSBr monolayer and the fundamental physical principles that distinguish the out-of-plane piezoelectricity between different CrXSYBrZ monolayers. Importantly, combined with the electronegativity difference effect, we revealed the strong coupling mechanism between ferromagnetism and out-of-plane piezoelectricity of CrXSYBrZ.

2. Computational details

We performed density functional theory (DFT) calculations using the Vienna ab initio simulation package (VASP).³⁰ The exchange–correlation effect is described by the generalized gradient approximation (GGA) using the functional proposed by Perdew, Burke, and Ernzerhof (PBE).³¹ For CrSBr and CrXSYBrZ monolayers, the vacuum spaces between adjacent slabs are set at least to be 15 Å. For relaxed structures, the force of each atom in the unit cell is lower than 10⁻³ eV Å⁻¹. The change of the total energy is lower than 10⁻⁶ eV. The Monkhorst–Pack grid of 12 × 12 × 1 k-mesh is used for CrSBr and CrXSYBrZ during structural relaxations. For CrSBr, the optimized lattice constants 3.54 Å and b = 4.73 Å are almost the same as the experimental bulk value and the same as the previous DFT calculations value. We use 4 × 4 × 1 supercell to calculate phono spectra using PHONOPY software interfaced with VASP.³² The kinetic energy cutoff for plane-wave expansion is set to 450 eV. To describe the correlation effect of Cr 3d electrons and Mo 4d electrons, the GGA plus Hubbard U (GGA+U) is employed, with the common values of U = 4 eV (for Cr) and 3 eV (for Mo) and Hund exchange J_H = 1 eV (and the effective U_eff = U − J_H = 3 eV (for Cr) and 2 eV (for Mo)).^19,33 To study different magnetic structures, 2 × 2 × 1 supercells for these monolayers were used in our calculation. The spin–orbit-coupling is included in our calculations on magnetism to evaluate the magnetic anisotropy. The Heyd–Scuseria–Ernzerhof (HSE06) functional of the generalized Kohn–Sham scheme is used to obtain the electronic band structures.³⁴ For CrSBr, the band gap is calculated to be 1.66 eV, which is in the range of experimental value (1.5 ± 0.2 eV).³⁵ Density functional perturbation theory (DFPT) is used to calculate the phono spectra, elastic stiffness tensors C_ij and piezoelectric coefficients e_il.

3. Results and discussion

3.1 Optimized CrXSYBrZ monolayers and stability

The process of atomic substitutions in the CrSBr monolayer to generate a series of CrXSYBrZ monolayers is clearly illustrated in the side views (crystal planes (100) and (010)) presented in Fig. 1. In Fig. 1(b), the replacement of the first layer of Br atoms with Z atoms along the z-axis, from the top to the bottom of CrSBr, leads to the formation of three Cr₂S₂BrZ monolayers. Similarly, substituting the second layer Cr atoms with Mo and the third layer S atoms with Se atoms results in the creation of Cr₂SSeBr₂ and CrMoS₂Br₂ monolayers, respectively (as shown in Fig. 1(c) and (d)). The structural parameters of CrSBr and CrXSYBrZ are detailed in Table 1. After the complete relaxation of all atoms, the lattice constants of CrSBr (a = 3.54 Å and b = 4.73 Å) closely align with the experimentally verified bulk CrSBr (a = 3.50 Å and b = 4.76 Å)³⁵ and are consistent with those from prior theoretical investigations,³⁶ showcasing the precision of our first-principles calculations. The smaller atomic radii of the Z atoms (H: 32 pm, F: 64 pm and Cl: 99 pm) in Cr₂S₂ZBr, in comparison to Br atoms they replace (114 pm), lead to slightly reduced lattice constants a for Cr₂S₂ZBr (Cr₂S₂BrH: 3.34 Å, Cr₂S₂BrF: 3.38 Å and Cr₂S₂BrCl: 3.50 Å) when compared to CrSBr. However, as the radii of Z atoms are all smaller than those of the Cr (118 pm) and S (104 pm) atoms, the lattice constant b of each Cr₂S₂ZBr remains almost nearly unchanged in comparison to CrSBr. Similarly, the larger atomic radii of Se (117 pm) and Mo (130 pm) atoms, in contrast to S (104 pm) and Cr (118 pm) atoms, results in both lattice constants of Cr₂SSeBr₂ (a = 3.59 Å and b = 4.90 Å) and CrMoS₂Br₂ (a = 3.63 Å and b = 4.85 Å) being larger than those of CrSBr. Additionally, the size relationship of these atoms’ radii (r) is clearly evident in the thicknesses (as shown in Table 1) relationship of these six monolayers: r(Mo) > r(Cr) > r(Se) > r(Br) > r(S) > r(Cl) > r(F) > r(H) corresponds to h(CrMoS₂Br₂) > h(Cr₂SSeBr₂) > h(CrSBr) > h(Cr₂S₂BrCl) > h(Cr₂S₂BrF) > h(Cr₂S₂BrH).

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As depicted in Fig. 2(a)–(f), the phonon spectra of CrSBr and CrXSYBrZ monolayers do not exhibit any imaginary frequencies, indicating that they contain great dynamic stability. The formation energy E_f is also calculated to assess the feasibility of experimental synthesis, which of CrSBr and CrXSYBrZ can be calculated by:³⁷

Ef = Etotal − (nX × EX + nY × EY + nZ × EZ)/n,	(1)

where E_total represents the total energy of CrSBr and CrXSYBrZ, n_X/Y/Z and E_X/Y/Z denote the number and chemical potential of the X/Y/Z atom, respectively.³⁸ And n (= 6) represents the atomic number in the unit cell of these monolayers. The formation energy data is presented in Table 1. As illustrated in Fig. 2(g), the formation energy values of these monolayers are all negative, ranging from the lowest value of Cr₂S₂BrF (−1.32 eV per atom) to the highest value of CrMoS₂Br₂ (−0.74 eV per atom). Intriguingly, the formation energies of Cr₂S₂ZBr are all lower than that of the initial structure CrSBr (−0.99 eV per atom) that has already been experimentally exfoliated, which proved that Cr₂S₂ZBr exhibit excellent thermodynamic stability. Exfoliation energy is used to evaluate the difficulty of monolayer exfoliation from its corresponding bulk. To further prove the feasibility of the future experimental synthesis of CrXSYBrZ monolayers, we calculated the exfoliation energy of these monolayers using the following formula:

	(2)

where E_2D and E_3D respectively indicate the total energy of the monolayer and bulk states of the same materials. S_3D indicates the base area of the bulk material. In Table 1, the exfoliation energy of Cr₂S₂BrH (25.51 meV Å⁻²) is the minimum value among CrXSYBrZ monolayers, which even approaches the exfoliation energy of graphene (21 meV Å⁻²).³⁹ It proves that it takes less energy to exfoliate CrXSBrZ monolayers out of their corresponding bulk materials and provides an advantage for the future experimental synthesis of these monolayers.

The elastic stiffness coefficients C_ij, Young's modulus, and Poisson's ratio data for CrSBr and CrXSYBrZ monolayers at θ = 0° (x-direction) and θ = 90° (y-direction) are presented in Table 2. The C_ij values of these monolayers all meet the Born–Huang criteria:⁴⁰ C₁₁C₂₂ − C₁₂² > 0 and C₆₆ > 0, indicating their mechanical stability. In the case of 2D materials, Young's modulus Y(θ) reflects the material's resistance to deformation along the direction of an applied force. It quantifies the material's deformation under stress and is a crucial parameter for understanding the material's mechanical behavior. The Poisson's ratio v(θ) of 2D materials reveals how the material deforms laterally when subjected to an axial load. The Y(θ) and v(θ) of CrSBr and CrXSYBrZ can be expressed as follows:⁴¹

	(3)

	(4)

where θ represents an arbitrary angle with respect to the x-direction, and H = (C₁₁C₂₂ − C₁₂²)/C₆₆. As shown in Fig. 2(h), CrSBr and CrXSYBrZ demonstrate in-plane anisotropy in Y(θ) and v(θ). Their Y(θ) reaches the relatively larger value at 0° (Y_a) and 90° (Y_b), while v(θ) attains their relatively smaller value at these angles. These characteristics stem from the variations in the lattices of each monolayer along the x and y directions. Due to atomic substitutions, the Y_a of CrXSYBrZ (ranging from 71.78 to 97.72 N m⁻¹) are consistently higher than that of CrSBr (64.82 N m⁻¹). In Cr₂S₂BrH and Cr₂S₂BrF, the Y_b values (107.05 and 106.51 N m⁻¹) surpass those of the other CrXSYBrZ monolayers, attributed to the stronger bonding of H and F atoms with their neighboring Cr atoms. CrSBr and CrXSYBrZ exhibited positive Poisson's ratio (ranging from 0.03 to 0.21), contracted laterally under tension and expanded laterally under compression. In essence, Cr₂S₂BrH and Cr₂S₂BrF demonstrate increased hardness along the x and y directions due to their higher Y_a and Y_b. Cr₂SSeBr₂ and CrMoS₂Br₂ exhibited relatively elevated v values (v_a = 0.15 and v_b = 0.21), rendering them more likely to longitudinal changes under in-plane stress. Additionally, both Y_a and Y_b values of CrSBr and CrXSYBrZ are lower than that of graphene (330 N m⁻¹),⁴² which indicates that the exceptional softness and flexibility of these six monolayers as promising 2D materials with great potential for wearable device applications.

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3.2 Ferromagnetism and electronic properties of CrXSYBrZ monolayers

In order to accurately calculate and not underestimate the effects of the d-orbital electrons of Cr and Mo atoms on the electronic structure and magnetic moment, we employed GGA+U calculations when computing the electronic band structures of these monolayers. In previous studies, both CrSBr (with Curie temperature T_c of 175 K) and CrMoS₂Br₂ (T_c = 360 K) have been identified as ferromagnetic monolayers.^33,43 As illustrated in Table S2 (see the ESI†), we computed the energy differences ΔE of these monolayers in both ferromagnetic and antiferromagnetic states compared to the total energy without considering magnetism. The total energies of these monolayers in the ferromagnetic state are lower than those without magnetism, which is indicated by the negative values of ΔE. In contrast, the total energies in the antiferromagnetic state are higher than those without magnetism, which results in the positive values of ΔE. It suggests that the ferromagnetic state is the ground state of these monolayers. This phenomenon is attributed to the half-filled electronic states of the Cr 3d orbitals and Mo 4d orbitals in these monolayers. Notably, the magnetic moment of Cr atoms in CrMoS₂Br₂ is 3.40μ_B (as shown in Table S2, ESI†), while that of Mo atoms is 2.65μ_B, indicating that Cr atoms exhibit stronger magnetism compared to Mo atoms in CrMoS₂Br₂. This difference is attributed to the distinct atomic structures of Cr and Mo atoms, resulting in various degrees of hybridization in their d orbitals. The hybridization effect influences the spin arrangement of electrons and the magnitude of the magnetic moment. Herein, we used the ferromagnetic state and three antiferromagnetic state configurations of the CrSBr monolayer shown in Fig. 3 in ref. 36 to calculate three intralayer exchange parameters J₁, J₂ and J₃ of Cr₂S₂BrZ and Cr₂SSeBr₂ monolayers, which can be seen in Fig. 1 (a) of ref. 19. The J₁, J₂ and J₃ data are listed in Table S3 (see the ESI†). In Fig. 3, T_c of Cr₂S₂BrZ and Cr₂SSeBr₂ monolayers vary from the minimum of 250 K in Cr₂S₂BrF and Cr₂S₂BrCl to the maximum of 340 K in Cr₂S₂BrH are larger than that of CrSBr monolayer. This indicates that the substitutions of atoms X, Y and Z in CrSBr induce the rise of T_c.

The electronic band structures and orbital-resolved density of states of CrSBr and CrXSYBrZ are illustrated in Fig. 4. Among these six direct semiconductors, the bandgap spans from 1.08 eV (Cr₂SSeBr₂) to 1.70 eV (Cr₂S₂BrCl), which manifests these monolayers are narrow bandgap semiconductors.⁴⁴ The projected density of states for each monolayer in Fig. 4 reveals that the conduction band minimum (CBM) is primarily influenced by the d-shell electrons of the Cr atom, while the valence band maximum (VBM) is primarily influenced by the p-shell electrons of the Br atom. The atomic substitution processes from CrSBr to CrXSYBrZ involve the replacement of individual Cr or Br atoms rather than all at once, ensuring that the CBM and VBM positions of these six structures remain at the G point. This preservation maintains the direct semiconductor characteristics of CrXSYBrZ. Besides, we consider the influence of the spin–orbit coupling effect (SOC) on the electric band structures of CrXSYBrZ monolayers under GGA+U calculation. As shown in Table S1 and Fig. S1 (see the ESI†), SOC did not change the CBM and VBM positions of these monolayers, but greatly reduced the band gaps of these monolayers except Cr₂S₂BrH, which is because the H atom has a smaller atomic mass. Considering SOC, the band gaps of Cr₂S₂BrCl and Cr₂S₂BrF decrease by 0.46 and 0.37 eV, which are larger values among that of CrXSYBrZ monolayers. Halogen elements F and Cl are more electronegative than Br, and replacing one of the Br atoms in CrSBr with an F or Cl atom causes more active charge transfer in the corresponding monolayer, making SOC more likely to occur.

From Fig. 4, it can be observed that the CBM along the path from the high-symmetry point G to X in CrSBr and CrXSYBrZ exhibit an ultra-flat electronic band.⁴⁵ Previous studies have reported that flat electronic bands can lead to a series of quantum phases driven by interactions, including ferromagnetism,⁴⁶ Mott insulating phases induced by electron correlations,⁴⁷ and superconductivity.⁴⁸ In Fig. 4, the projection of the density of states for these monolayers indicates that these flat electronic bands are primarily contributed by the 3d orbital electrons of the Cr atoms, which also plays a decisive role in the ferromagnetism of these monolayers. Therefore, it may be inferred that these flat bands have a strong relationship with the ferromagnetism of these monolayers.

Table 3 presents the exact values of the Fermi energy E_F, vacuum level V_L, VBM, CBM and work function Φ of CrSBr and CrXSYBrZ monolayers. The substitution of Mo atoms, or more specifically, the replacement of Cr atoms, significantly elevates the position of the E_F and also raises the V_L position. It once again reflects the significance of the electrons of Cr atoms on the electronic structure of these monolayers. The work function Φ is a crucial electron characteristic that indicates their ability to escape from a material's surface.⁴⁹ The Φ can be calculated using the V_L and the E_F:

Φ = VL − EF,	(5)

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The Φ of these six monolayers ranges from a minimum of 6.50 eV for Cr₂S₂BrH to a maximum of 7.09 eV for Cr₂S₂BrF, all of which exceed graphene's 4.6 eV.⁵⁰ The higher work functions of these monolayers can be attributed to their strong symmetries (C_2v and D_2h) and the presence of filled d orbitals in the Cr and Mo atoms within them.⁵¹ It indicates that in CrSBr and CrXSYBrZ, electrons require a greater amount of energy to escape from the material surface.

The vertical mirror symmetry D_2h of CrSBr is clearly evident in its electronic properties from the planar-averaged charge density with Bader charges analysis and electrostatic potential along the z-axis. In Fig. 5(a), the identical atomic species in the upper and lower three layers along the z-axis in CrSBr, along with their symmetric distribution relative to the (001) crystal plane, result in two symmetry patterns in the plane-averaged charge distribution and electrostatic potential curve along the z-axis in CrSBr (as shown in Fig. 5(a)). This symmetry is further emphasized by the equal electrostatic potential values on the two opposing surfaces along the z-axis in CrSBr, indicating the absence of intrinsic polarization along the z-direction in CrSBr. Moreover, as shown in Table 4, the Bader charge values of the two Cr atoms (1.19 |e|), two S atoms (−0.76 |e|), and two Br atoms (−0.44 |e|) in CrSBr are identical, demonstrating the complete symmetry of CrSBr. This can be embodied on that the charge transfer in CrSBr is symmetric. The symmetry of CrXSYBrZ, C_2v, loses mirror symmetry compared to CrSBr, which is also shown in Fig. 5(b)–(f). The substitution of the X, Y and Z atoms leads to a non-symmetric plane-averaged charge distribution and electrostatic potential curves in CrXSYBrZ. Furthermore, each CrXSYBrZ introduces an additional splitting of the vacuum energy level along the z-direction, with the difference in electrostatic potential ΔΦ varying from 0.12 eV (Cr₂SSeBr₂) to 0.56 eV (Cr₂S₂BrF). This shift indicates the intrinsic polarizations along the z-direction in CrXSYBrZ, which leads to the out-of-plane piezoelectricity of these monolayers. Additionally, the Bader charge values of each atom in CrXSYBrZ no longer exhibit symmetric distribution as observed in CrSBr.

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The electronegativity relationships of the atoms in these monolayers are effectively demonstrated through the Bader charge analysis.⁵² Taking Cr₂S₂BrH as an example, the electronegativity numerical relationship of the atoms is: Br (2.96) > S (2.58) > H (2.1) > Cr (1.66). Given that Cr atoms possess the lowest electronegativity value and are situated between the H, S, and Br atoms, as indicated in Table 4, two Cr atoms exhibit positive Bader charge values (1.16 |e|) due to electron loss, which is transferred to the neighboring H, S, and Br atoms. Consequently, the Bader charge values of the H, S and Br atoms are negative. It is noteworthy that among these monolayers, the electronegativity of Br atoms surpasses that of S atoms. However, Br atoms attract fewer electrons from their neighboring Cr atoms compared to S atoms. This disparity arises from the positions of two S atoms between two Cr atoms, while Br atoms are situated externally to the Cr atoms. As shown in Fig. 5, the number of bonds between Cr atoms and S (or Se) atoms is double that between Cr atoms and Br atoms. This phenomenon is further evidenced in the charge density difference depicted in Fig. 5, where the yellow region denoting charge accumulation between Cr atoms and S (or Se) atoms is more extensive than the corresponding region between Cr atoms and Br atoms within the same structure.

It is noteworthy that we conducted GGA+U to calculate the Bader charge values of these monolayers. Taking Cr₂S₂BrH for an instance, the Bader charge values of the two Cr atoms are both 1.16 |e| without considering the U parameter. After considering the U parameter, the Bader charge values of Cr1 and Cr2 atoms increased to 1.21 and 1.23 |e|, respectively (in Table 4). Here, the numbering of the Cr atoms 1 and 2 corresponds to the atom with larger and smaller z-axis coordinates, respectively. This trend is also consistent across the other five monolayers. GGA+U calculations accurately capture the impact of the Cr and Mo d-shell atomic magnetic moments on the electronic structure. Compared with GGA calculation, GGA+U results in a larger atomic magnetic moment, which causes Cr and Mo atoms to lose more electrons in these monolayers.

3.3 Carrier properties of CrXSYBrZ monolayers

As depicted in Fig. 6(a) and (d), the total energy of CrSBr and CrXSYBrZ monolayers exhibit parabolic variations with x/y uniaxial strain. Utilizing formula (S12) (see the ESI†), the elastic stiffness coefficient C_2D along the x/y direction of these monolayers can be calculated. The band-edge positions of these monolayers demonstrate linear variations with x/y uniaxial strain as shown in Fig. 6(b) and (e). Through formula (S13) (ESI†), the deformation potential constants E_d along the x/y direction of these monolayers can be computed. The values for C_2D, E_d, and the effective mass m* of carriers for CrSBr and CrXSYBrZ are detailed in Table 5. The differences in the lattice constants along the x and y directions of these monolayers lead to various carrier effective masses along these directions. For instance, the electron effective mass of Cr₂S₂BrH is 1.94/2.76 m_e along the x/y direction. The electron effective masses of CrSBr (34.64 m_e), Cr₂S₂BrCl (42.33 m_e), Cr₂SSeBr₂ (31.75 m_e), and CrMoS₂Br₂ (42.33 m_e), and the hole effective mass of Cr₂S₂BrF (14.65 m_e) along the x-direction, exhibit significantly high values. According to formula (S11) (ESI†), the inverse of the carrier effective mass shows a positive correlation with the second derivative of the band edge position concerning the k-point coordinates. Based on the observation of the flat band phenomenon in the electronic band structures of these monolayers shown in Fig. 4, it can be deduced that in these flat band regions, the band edge position remains nearly constant with respect to the k-point coordinates. This leads to the second derivative term mentioned earlier being zero, indicating that the inverse of the carrier effective mass is zero, which effectively translates to infinity. Consequently, these monolayers demonstrate significantly large carrier masses. Moreover, the strong correlation between the flat band phenomenon and the ferromagnetism in these monolayers suggests that the ferromagnetic properties will enhance the effective mass of carriers within them. In these monolayers, the electron mobility in the x-direction of Cr₂S₂BrCl stands out as the highest, reaching an impressive 8.41 × 10³ cm² V⁻¹ s⁻¹. This excellent performance is attributed to its VBM's remarkable insensitivity to x-direction uniaxial strain. Furthermore, the electron carrier mobilities of Cr₂S₂BrF along both x (0.31 × 10³ cm² V⁻¹ s⁻¹) and y-directions (0.61 × 10³ cm² V⁻¹ s⁻¹) are notably elevated, surpassing the mobility along the y-direction of MoS₂ monolayer (0.20 × 10³ cm² V⁻¹ s⁻¹)⁵³ by 1.55 and 3.05 times, respectively. It suggests that these 2D materials hold great potential in high-speed nanoelectronics. As depicted in Fig. 6(c) and (f), the atomic magnetic moments of Cr and Mo atoms in CrSBr and CrXSYBrZ exhibit linear increase with x/y uniaxial strain. It implies that the magnetic moments of Cr and Mo atoms in these monolayers can be manipulated under strain, thus these 2D materials are desirable candidates for magnetic storage devices.

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3.4 Piezoelectricity of CrXSYBrZ monolayers

From Table 6, the out-of-plane stress and strain piezoelectric coefficients are detailed. The presence of non-zero piezoelectric coefficients signifies the exhibition of out-of-plane piezoelectricity in CrXSYBrZ monolayers. Among these monolayers, Cr₂S₂BrF stands out with the largest out-of-plane stress piezoelectric coefficients: e₃₁ = −1.34 × 10⁻¹⁰ C m⁻¹, e₃₂ = −1.14 × 10⁻¹⁰ C m⁻¹, and e₃₃ = −1.53 × 10⁻¹⁰ C m⁻¹. Additionally, Cr₂S₂BrF demonstrates the stiffest elastic properties, reflected in its largest Y(θ) value (refer to Fig. 3(h)). Utilizing formulas (S7)–(S9) (ESI†), Cr₂S₂BrF exhibits the highest out-of-plane strain coefficients among these monolayers: d₃₁ = −1.57 × 10⁻¹⁰ C m⁻¹, d₃₂ = −1.82 × 10⁻¹⁰ C m⁻¹, and d₃₃ = 20.50 × 10⁻¹⁰ C m⁻¹. The d₃₃ of Cr₂S₂BrF is 200 times larger than that of MoSSe monolayer (d₃₃ = 0.1 pm V⁻¹)^54,55 known for its exceptional out-of-plane piezoelectricity.

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The large electronegativity difference ratio is evident in the enhanced out-of-plane piezoelectricity along the 33 direction of a 2D material composed of three atomic layers.^38,56,57 Among the MoTO (T = S, Se or Te) monolayers, the ratio of electronegativity difference between T and O atoms and that between T and Mo atoms is the largest in MoTeO, so it has the largest out-of-plane piezoelectric coefficient e₃₃.³⁸ The physical mechanism behind this phenomenon is the asymmetric electron transfer between atoms within 2D material. The intensity of asymmetric electron transfer inside materials promotes more charges to move to the material surface when subjected to strain/stress, consequently enhancing the out-of-plane piezoelectricity of materials. Besides, Guo et al. proposed the electronegativity difference concept and well explained that the out-of-plane piezoelectric coefficient d₃₁ of CrYX (Y = S; X = Cl, Br, I) and CrYX (Y = O; X = F, Cl, Br) is positively related to electronegativity difference of X and Y atoms.⁵⁸ Based on the electronegativity difference ratio and electronegativity difference concepts, we partitioned CrSBr and CrXSYBrZ monolayers along the z-axis into two parts: T (consisting of the upper three atomic layers along the z-axis) and B (consisting of the lower three atomic layers along the z-axis). The electronegativity difference R between the parts T and B can be expressed as:

R = |ET − EB|,	(6)

where E_T and E_B respectively represent the sum of atomic electronegativity within the parts T and B, which are denoted as the electronegativity of parts T and B. R reflects the intensity of the asymmetric charge transfer between parts T and B in these monolayers. As depicted in Fig. 7(a), the |e₃₃| values of these monolayers increase with the electronegativity difference between parts T and B in them. Given that the CrSBr and CrXSYBrZ comprise six atomic layers, we term this phenomenon as the electronegativity difference effect in CrSBr and CrXSYBrZ. In CrSBr, the atomic composition within parts T and B are identical, resulting in E_T = E_B = E_Cr + E_S + E_Br, where E_Cr, E_S and E_Br represent the electronegativity of Cr, S and Br atoms, respectively. Therefore, the electronegativity difference R in CrSBr is zero, leading to the symmetric charge transfer in each part and no charge transfer between the parts T and B. Consequently, CrSBr lacks out-of-plane piezoelectricity. This observation is corroborated in Table 4, where the Cr atoms within parts T and B in CrSBr equally lost the same electrons to the S and Br atoms in each part. Among CrXSYBrZ, the asymmetric charge transfer occurs most intensely in Cr₂S₂BrF with R = 1.02, thus exhibiting the best out-of-plane piezoelectricity. Notably, the electronegativity difference R in CrXSYBrZ stems solely from the variation in electronegativity of the two atoms pre and post-substitution. For instance, in Cr₂S₂BrZ, the electronegativity difference between parts T and B is equivalent to that between the Z and Br atoms.

To explore the coupling mechanism of ferromagnetism on the out-of-plane piezoelectricity of CrXSYBrZ monolayers, we conducted GGA+U calculation of the out-of-plane piezoelectric coefficient e₃₃ of these monolayers (the corresponding data are detailed in Table S4, ESI†), which is used to avoid underestimating the atomic magnetic moment. The influence of the enhanced ferromagnetism on the e₃₃ values of other CrXSYBrZ is negligible except for CrMoS₂Br₂. In CrMoS₂Br₂, the e₃₃ value undergoes a significant change of 0.15 × 10⁻¹⁰ C m⁻¹ under GGA+U calculation, representing a relative decrease of −23.4% compared to the calculation without considering the U parameter due to the enhanced ferromagnetism. The electrons contribution to e₃₃ in CrMoS₂Br₂ reduced from 0.64 × 10⁻¹⁰ to 0.50 × 10⁻¹⁰ C m⁻¹ (as shown in Table S4, ESI†). The Bader charge analysis manifests that the U parameter leads Cr and Mo atoms to lose more electrons in these monolayers. In other CrXSYBrZ except CrMoS₂Br₂, each part T/B contains one Cr atom. Therefore, even though the U parameter affects Cr atoms, the intensity of the asymmetric charge transfer between parts T and B is not influenced. However, in CrMoS₂Br₂, the intensity of asymmetric charge transfer is precisely caused by that between Cr and Mo atoms. As depicted in Table 4, the inclusion of the U parameter in CrMoS₂Br₂ results in additional losses of 0.03 |e| for the Cr atom and 0.05 |e| for the Mo atom. Notably, the Mo atom loses 0.02 |e| more electrons than the Cr atom, and the Mo atom with a larger electronegativity (2.16) is more likely to lose electrons than Cr with a smaller electronegativity (1.66). Therefore, ferromagnetism actually shortens the difference between the ability of Cr and Mo atoms to lose electrons in CrMoS₂Br₂. Consequently, the intensity of the asymmetric charge transfer in CrMoS₂Br₂ is actually weakened due to the enhanced ferromagnetism, and thereby significantly decreases the e₃₃ of CrMoS₂Br₂.

We applied equal percentages of strain along the x and y axes simultaneously and allowed the internal atoms to relax, thus achieving the z-axial strain to CrXSYBrZ monolayers. As illustrated in Fig. 7(c), the out-of-plane piezoelectric coefficient e₃₃ values of these monolayers exhibit an approximately linear change with biaxial strain, suggesting that their out-of-plane piezoelectric properties can be manipulated through strain. Among these monolayers, the e₃₃ value of Cr₂S₂BrF has the largest variation range under the same strain range, reaching 0.14 × 10⁻¹⁰ C m⁻¹. These CrXSYBrZ monolayers are characterized by both flexibility and remarkable out-of-plane piezoelectricity, and have great potential in wearable nanoscale piezoelectric devices such as electronic skin and human health monitoring devices.^59,60

4. Conclusion

By replacing atomic layers in the CrSBr monolayer, we designed and predicted five stable 2D CrXSYBrZ semiconductors with ferromagnetism and excellent out-of-plane piezoelectricity. Among them, the Cr₂S₂BrF monolayer demonstrates the most outstanding out-of-plane piezoelectricity. The ferromagnetism in these monolayers leads to the large carrier effective masses based on the flat band regions in their electronic structures. These materials exhibit high carrier mobility. In particular, Cr₂S₂BrCl showcases a remarkable electron mobility of 8.41 × 10³ cm² V⁻¹ s⁻¹ along the x-direction. The electronegativity difference effect is defined as the e₃₃ value increasing with the electronegativity difference between the upper and lower parts in CrSBr and CrXSYBrZ. In CrMoS₂Br₂, the enhanced ferromagnetism increases the magnetic moments of Cr and Mo atoms and makes them lose more electrons, which weakens the electron contribution to e₃₃. Besides, the actual inside asymmetric charge transfer is less intense in CrMoS₂Br₂ due to the enhanced ferromagnetism, which in turn decreases the polarization charge distribution, thereby significantly decreasing the out-of-plane piezoelectricity of CrMoS₂Br₂. These monolayers exhibit internally strain-tunable magnetic moments of Cr and Mo atoms and out-of-plane piezoelectricity, indicating significant potential for nanoelectronics and nanomagnetic functional devices.

Data availability

Data for this paper are available through the corresponding author.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The work was funded by the National Natural Science Foundation of China (no. 11474123 and 11574116) and the Natural Science Foundation of Jilin Province of China (no. 20220101015JC).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc02409e

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