Anomalous valley Hall effect in electric-potential-difference antiferromagnetic $\mathrm{Cr_{2}CHCl}$ monolayer

The discovery and successful preparation of rich two-dimensional (2D) materials lays the foundation for valleytronics, which can process information and perform logic operations with low power consumption and high speedq1 ; q2 ; q3 ; q4 ; ref1 ; ref2 ; ref3 ; ref4 . Recently, the ferrovalley semiconductor (FVS) has been proposed to realize intrinsic valley polarizationq10 , which appears in out-of-plane hexagonal ferromagnetic (FM) materials with broken spatial inversion symmetryq10 ; q11 ; q12 ; q13 ; q13-1 ; q14 ; q15 ; q16 ; q17 ; q18 . These offer interesting platforms to study valley-contrasting transport and Berry physics. In the FVS, valley-dependent Berry curvature can produce anomalous valley Hall effect (AVHE), where the charge Hall current originates from the spontaneous valley polarization. The antiferromagnetic (AFM) materials possess the high storage density, robustness against external magnetic field, as well as the ultrafast writing speedv12 , so realizing valley polarization and AVHE in AFM materials is more meaningful for valleytronic application.

For 2D AFM materials with $PT$ symmetry (a combined symmetry of spatial inversion ($P$) and time reversal ($T$)), there is zero berry curvature ($\Omega(k)$) and no spin-splitting everywhere in the momentum space, which hinders the realization of AVHE. Recently, an intuitive way is proposed to produce spin-splitting in AFM materials by making the magnetic atoms with opposite spin polarization locating in the different environment (surrounding atomic arrangement)gsd . The altermagnetismk6 and electric-potential-difference antiferromagnetism (EPD-AFM)k7 are the representative examples, and they possess intrinsic spin-splitting. For altermagnetism, the two different environments (The surrounding atoms are in the same arrangement, yet not in the same orientation.) can be connected by special symmetry operationk6 . For EPD-AFM, the different environments occupied by two (spin-up and spin-down) magnetic atoms are due to an electric-potential-difference caused by an out-of-plane built-in electric field, and the magnetic atoms have opposite layer spin polarization (A-type AFM ordering). If EPD-AFM has hexagonal symmetry, the layer-locked Berry curvature can appeargsd1 . If hexagonal symmetry couples with the out-of-plane magnetization for EPD-AFM, spontaneous valley polarization will existgsd1 . Therefore, an out-of-plane hexagonal EPD-AFM with energy extrema of conduction or valance bands located at high symmetry -K and K points can achieve AVHE.

The experimentally synthesized $\mathrm{Cr_{2}C}$ is a half-metallic ferromagnetv16 . By surface functionalization with H or Cl in $\mathrm{Cr_{2}C}$, both ferromagnetic-antiferromagnetic transition and metal-insulator transition can be induced simultaneouslyv15 . The functionalized $\mathrm{Cr_{2}CH_{2}}$ and $\mathrm{Cr_{2}CCl_{2}}$ possess A-type AFM ordering and energy extrema at -K and K high symmetry points. However, for $\mathrm{Cr_{2}CH_{2}}$ and $\mathrm{Cr_{2}CCl_{2}}$, the spin degeneracy of -K and K valleys is maintained due to $PT$ symmetry, which prohibits the AVHE. Based on $\mathrm{Cr_{2}CH_{2}}$ and $\mathrm{Cr_{2}CCl_{2}}$, a Janus $\mathrm{Cr_{2}CHCl}$ can be constructed to achieve hexagonal EPD-AFM, which is a possibe candidate material to achieve AVHE.

Within density functional theory (DFT)1 , we perform the spin-polarized first-principles calculations within the projector augmented-wave (PAW) method, as implemented in Vienna ab initio Simulation Package (VASP)pv1 ; pv2 ; pv3 . We use the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE-GGA)pbe as the exchange-correlation functional. The kinetic energy cutoff of 500 eV, total energy convergence criterion of $10^{-8}$ eV, and force convergence criterion of 0.0001 $\mathrm{eV.{\AA}^{-1}}$ are adopted. To account for the localized nature of Cr-3$d$ orbitals, a Hubbard correction $U_{eff}$=3.0 eVv13 ; v13-1 is used by the rotationally invariant approach proposed by Dudarev et alu . The spin-orbital coupling (SOC) is included to investigate valley splitting and magnetic anisotropy energy (MAE). To avoid interactions between neighboring slabs, the vacuum space of more than 20 $\mathrm{{\AA}}$ along $z$ direction is added. We use a 21$\times$21$\times$1 Monkhorst-Pack k-point meshes to sample the Brillouin zone (BZ) for calculating electronic structures. Based on finite displacement method with 5$\times$5$\times$1 supercell, the interatomic force constants (IFCs) are calculated with AFM ordering, and the phonon dispersion is constructed by the Phonopy codepv5 . The ab initio molecular dynamics (AIMD) simulations using NVT ensemble are performed with a 4$\times$4$\times$1 supercell for more than 8000 fs with a time step of 1 fs. The Berry curvatures are calculated directly from the calculated wave functions based on Fukui’s methodbm , as implemented in the VASPBERRY codebm1 ; bm2 ; bm3 .

The $\mathrm{Cr_{2}CH_{2}}$ and $\mathrm{Cr_{2}CCl_{2}}$ monolayers have been proved to possess A-type AFM ordering with good stabilitiesv15 . Monolayer $\mathrm{Cr_{2}CHCl}$ can be constructed by replacing one of two H (Cl) layers with Cl (H) atoms in $\mathrm{Cr_{2}CCH_{2}}$ ($\mathrm{Cr_{2}CCl_{2}}$) monolayer. The crystal structures of $\mathrm{Cr_{2}CHCl}$ are shown in Figure 1 (a) and (b), which crystallizes in the $P3m1$ (No. 156), lacking spatial inversion symmetry. The $\mathrm{Cr_{2}CHCl}$ consists of five atomic layers in the sequence of H-Cr-C-Cr-Cl, which can produce built-in electric field due to special Janus structure. In addition, the magnetic Cr atoms of $\mathrm{Cr_{2}CHCl}$ distribute in two layers. These provide the basic conditions for EPD-AFM. The FM and three AFM configurations (AFM1, AFM2 and AFM3) are constructed, as shown in FIG.S1 of electronic supplementary information (ESI), to determine magnetic ground state of $\mathrm{Cr_{2}CHCl}$. The AFM1 ordering is called A-type AFM state, which is necessary to form EPD-AFM. It is found that the energy of AFM1 per unit cell is 462 meV, 419 meV and 636 meV lower than those of FM, AFM2 and AFM3 cases within GGA+$U$, confirming that the monolayer $\mathrm{Cr_{2}CHCl}$ possesses AFM1 ground state with the optimized equilibrium lattice constants $a$=$b$=3.09 $\mathrm{{\AA}}$. The calculated phonon spectrums of $\mathrm{Cr_{2}CHCl}$ show no obvious imaginary frequencies (see FIG.S2 of ESI), indicating its dynamic stability. Based on FIG.S3 of ESI, the AIMD simulation shows that the framework of $\mathrm{Cr_{2}CHCl}$ is well preserved with little fluctuations of total energy with increasing time during the simulation period, which confirms its thermal stability.

The magnetization direction is of great significance for generating spontaneous valley polarization, and only out-of-plane magnetization direction can produce spontaneous valley splittinggsd1 . The magnetic orientation can be determined by calculating MAE, which can be calculated by $E_{MAE}=E^{||}_{SOC}-E^{\perp}_{SOC}$. The $E^{||}_{SOC}$ and $E^{\perp}_{SOC}$ are the energies that spins lie in-plane and out-of-plane, respectively. The calculated MAE is 31$\mathrm{\mu eV}$/unit cell, and the positive value indicates the out-of-plane easy magnetization axis of $\mathrm{Cr_{2}CHCl}$, which means spontaneous valley polarization.

The energy band structures of $\mathrm{Cr_{2}CHCl}$ are plotted in Figure 1 (c) and (d) without SOC and with SOC for magnetization direction along the positive $z$ direction. According to Figure 1 (c), the obvious spin-splitting can be observed due to the broken $PT$ symmetry, and $\mathrm{Cr_{2}CHCl}$ is an indirect band gap semiconductor. For $\mathrm{Cr_{2}CHCl}$, there are -K and K valleys with energy degeneracy in the valence band, but the valence band maximum (VBM) is not at the -K/K high symmetry point. The spin-splitting of $\mathrm{Cr_{2}CHCl}$ stems from a layer-dependent electrostatic potential, which occurs across the entire momentum space, producing the $s$-wave symmetry of spin-splittingk7 . This is different from altermagnetism ($d$, $g$, $i$-wave symmetry of spin-splitting) with momentum dependent spin-splittingk6 . When SOC is included, the spontaneous valley polarization with the valley splitting of 53 meV ($\Delta E_{V}=E_{K}^{V}-E_{-K}^{V}$) can be observed (Figure 1 (d)), and the energy of K valley is higher than one of -K valley. For $\mathrm{Cr_{2}CHCl}$, the total magnetic moment per unit cell is strictly 0.00 $\mu_{B}$, and the magnetic moment of bottom/top Cr atom is 3.13 $\mu_{B}$/-3.06 $\mu_{B}$. For EPD-AFM, the two types of magnetic (up and down) atoms cannot be connected by mirror or rotation symmetry, and the absolute values of their magnetic moments are not strictly equal, which is different from altermagnetism with strictly equal magnetic moments for up and down atomsk6 .

The VBM of $\mathrm{Cr_{2}CHCl}$ is not at the -K/K high symmetry point, which is not conducive to implementation of AVHE. Strain is a very effective way to tune the position of energy extrema for 2D materialsgsd2 . Here, the biaxial strain is applied to make VBM of $\mathrm{Cr_{2}CHCl}$ become -K or K point. We use $a/a_{0}$ (0.96 to 1.04) to simulate the biaxial strain, and the $a/a_{0}$$<$1 ($a/a_{0}$$>$1) means the compressive (tensile) strain, where $a$ and $a_{0}$ are the strained and unstrained lattice constants. Firstly, the magnetic ground state of strained $\mathrm{Cr_{2}CHCl}$ is determined by calculating energy differences ($\Delta E$) between FM/AFM2/AFM3 and AFM1 states vs $a/a_{0}$, as shown in Figure 2 (a). Within considered strain range, the positive $\Delta E$ confirm the AFM1 ground state of strained $\mathrm{Cr_{2}CHCl}$.

The energy band structures of $\mathrm{Cr_{2}CHCl}$ at representative $a/a_{0}$ without SOC and with SOC (out-of-plane magnetization direction) are shown in FIG.S4 of ESI, and the enlarged figures of the valence band near the Fermi level are plotted in Figure 3. For valence band, the valley splitting as a function of $a/a_{0}$ are plotted in Figure 4. It is clearly seen that the -K and K valleys always exist in the valence band, and they are from spin-up channel. It is found that the compressive strain can make -K/K valley become VBM, and the critical point is approximately 0.984 for $a/a_{0}$. The valley splitting of larger than 51 meV is maintained within considered $a/a_{0}$ range, and no valley polarization transition is produced. This is different from Janus $\mathrm{GdClF}$ monolayergsd3 , where the valley polarization transition can be driven by biaxial strain.

To achieve spontaneous valley splitting, another key factor is out-of-plane magnetization of strained $\mathrm{Cr_{2}CHCl}$. The MAE as a function of $a/a_{0}$ is plotted in Figure 2 (b). It is found that compressive strain can induce the transition of magnetization direction, and the magnetization of $\mathrm{Cr_{2}CHCl}$ changes from out-of-plane case to in-plane case, when the $a/a_{0}$ is lower than 0.968. By considering the MAE and position of VBM together, $\mathrm{Cr_{2}CHCl}$ is suitable to produce AVHE, when the $a/a_{0}$ is between 0.968 and 0.984.

Taking $a/a_{0}$=0.97 as a example, the energy band structures of valence bands near the Fermi energy level without SOC and with SOC for magnetization direction along the positive $z$, negative $z$, and positive $x$ directions are shown in Figure 5 (a), (b), (c) and (d). Figure 5 (a) shows obvious spin-splitting, and the VBM is at -K/K valley form the spin-up channel. However, no valley splitting can be observed. Figure 5 (b) shows the the valley splitting of 52 meV, which is higher than the thermal energy of room temperature (25 meV). The energy of K valley is higher than one of -K valley. Figure 5 (c) indicates that the valley polarization transition can be achieved by reversing the magnetization direction. Figure 5 (d) shows that the in-plane magnetization direction produces no valley polarization.

The SOC-induced valley splitting is mainly from the intra-atomic interaction $\hat{H}^{0}_{SOC}$ (the interaction between the same spin states)q18 :

in which $\hat{S}$, $\hat{L}$ and $\theta$/$\phi$ mean the spin angular momentum, orbital angular momentum and the polar angles of spin orientation, respectively. For $d_{x^{2}-y^{2}}$/$d_{xy}$-dominated -K/K valley with the group symmetry of $C_{3h}$, the valley splitting $\Delta E_{V}=4\alpha cos\theta$q18 , where the $\alpha$ is $\lambda|\hat{S}_{z}|$, and the $\theta$=0/90^∘ means out-of-plane/in-plane direction. For $\mathrm{Cr_{2}CHCl}$ with $a/a_{0}$=0.97, the Cr-$d$-orbital characters of energy bands within SOC are plotted in Figure 6, which shows $d_{x^{2}-y^{2}}$/$d_{xy}$-orbital-dominated -K and K valleys of valence bands. When the magnetization direction is along out-of-plane/in-plane case, the valley splitting of $\mathrm{Cr_{2}CHCl}$ will be 4$\alpha$/0.

Due to A-type AFM ordering in $\mathrm{Cr_{2}CHCl}$, the layer-locked Berry curvature can be observedgsd1 . Because -K and K valleys are from spin-up channel, the distribution of Berry curvatures of spin-up is shown in Figure 5 (e). It is clearly seen that the Berry curvatures are opposite for -K and K valleys. By applying a longitudinal in-plane electric field, the Bloch carriers will acquire an anomalous transverse velocity $v_{\bot}$$\sim$$E_{\parallel}\times\Omega(k)$v17 . When the Fermi level is shifted between the -K and K valleys of the valence band, only the spin-up holes of K valley move to the bottom boundary of the sample under an in-plane electric field (Figure 5 (f)), producing layer-locked AVHE. This accumulation of spin-polarized holes produces a net charge/spin current, which will generate observable voltage.

In summary, we present a hexagonal AFM monolayer $\mathrm{Cr_{2}CHCl}$ with spontaneous spin-splitting to realize AVHE. The spontaneous valley polarization can occur in $\mathrm{Cr_{2}CHCl}$ with the valley splitting of larger than 51 meV due to intrinsic out-of-plane magnetization, when the $a/a_{0}$ is larger than 0.968. The introduction of a built-in electric field caused by Janus structure induces the spin-splitting in monolayer $\mathrm{Cr_{2}CHCl}$ due to layer-dependent electrostatic potential. By combining with layer-locked Berry curvature, the layer-locked AVHE can be achieved in monolayer $\mathrm{Cr_{2}CHCl}$ without applying out-of-plane external electric field. Our works enrich AFM valleytronic materials, and provide advantageous for the development of energy-efficient and ultrafast electronic devices.