{'fraction_non_alphanumeric': 0.03986500738240877, 'fraction_numerical': 0.020248892638683822, 'mean_word_length': 4.2044456641053785, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Neel vector control in antiferromagnetic CrSb

CrSb is a layered antiferromagnet (AFM) with perpendicular magnetic anisotropy, a high Neel temperature, and large spin-orbit coupling (SOC), which makes it interesting for AFM spintronic applications. To elucidate the various mechanisms of Neel vector control, the effects of strain, band filling, and electric field on the magnetic anisotropy energy (MAE) of bulk and thin-film CrSb are determined and analysed using density functional theory. The MAE of the bulk crystal is large (1.2 meV per unit cell). Truncation of the bulk crystal to a thin film consisting of an integer number of unit cells breaks inversion symmetry, creates a large charge dipole and average electric field across the film, and breaks spin degeneracy, such that the thin film becomes a ferrimagnet. The MAE is reduced, and its sign can be switched with realistic strain. The large SOC gives rise to an intrinsic voltage controlled magnetic anisotropy (VCMA), with a VCMA coefficient similar to that of FeRh/MgO and other heavy-metal/ferromagnetic/MgO structures.


I. INTRODUCTION
Antiferromagnetic (AFM) materials are of great interest for future spintronics applications 1 . Their resonanat frequencies are much higher than those of ferromagnetic (FM) materials, which allows them to be used in the THz applications 2-4 and ultrafast switching 5 . However, it is challenging to control and detect the antiferromagnetic states. There are several methods to control the spins in AFMs such as via exchange bias with a proximity FM layer 6 and the use of electrical current by Néel spin-orbit torque 7 . The latter method has been extensively studied, although the results have recently been questioned 8,9 . Controlling the Néel vector without electrical current is promising for ultra low power applications, since it has been predicted that magnetization reversal can be achieved with aJ level energy consumption 10 . Electric field control of the magnetic properties of AFMs can be realized indirectly through the mechanism of mechanical strain created from a piezoelectric substrate [11][12][13][14] or a combination of strain plus exchange spring 15 . It can also be realized directly through the mechanism of voltage controlled magnetic anisotropy (VCMA). This mechanism has been experimentally and theoretically studied for FMs [16][17][18][19][20][21][22][23][24][25] , and the experimental results have been recently reviewed. 26 Technological applications have been described and analysed [27][28][29] . More recently, several theoretical studies of VCMA of AFMs have been reported 5,30,31 .
CrSb crystallizes in the hexagonal NiAs-type structure, and the spins on the Cr atoms couple ferromagnetically within the hexagonal plane and antiferromagnetically along the hexagonal axis as shown in Fig. 1. In the ground state, the Néel vector aligns along the hexagonal axis ([0001] direction), so that it has perpendicular magnetic anisotropy (PMA). The bands near the Fermi energy are composed of the d-orbitals of the Cr atoms, and these bands give rise to a large peak in the density of states near the Fermi energy. [33][34][35][36] The Sb atoms pro-vide significant SOC. CrSb has a high Néel temperature (705 K) making it suitable for on-chip applications 32 . Recently, CrSb has been used to control the magentic textures and tune the surface states of topological insulators [37][38][39] .
We examine three different mechanisms that alter the magnetic anisotropy of bulk and thin-film CrSb: (i) strain, (ii) electron filling, and (iii) electric field. Density functional theory (DFT) calculations of the magnetostriction coefficient, strain coefficient, filling coefficient, and VCMA coefficient characterize the effectiveness of the three methods in modifying the MAE.

II. METHODS
We perform first principles calculations as implemented in the Vienna Ab initio Simulation Package (VASP) 40 to investigate the effects of strain, electric field, and band filling on the magnetic anisotropy of CrSb. Projector augmented-wave (PAW) potentials 41 and the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) are employed 42 . A cut-off energy of 500 eV and 8 × 8 × 8 k-points grid are used to make sure the total energy converged within 10 −7 eV per unit cell. A GGA+U implementation is also used to reproduce the magnetic moment on the Cr atom corresponding to the experimental values 43 . The numerical value is U ef f = 0.25 eV for the Cr atom where From the initial bulk structure, uniaxial strain along the x-axis is applied and the structure is fully relaxed along y and z axes until all forces on each atom are less than 10 −3 eVÅ −1 . Here, x, y, and z axes are the ones from the Cartesian coordinate system, and the x and z axes are parallel to the a and c axes in the Fig. 1, respectively, since the unit cell is hexagonal. The strain is defined as ε = (a − a 0 )/a 0 × 100% where a and a 0 are the lattice constants along x with and without strain, respectivley. The calculated lattice constants without strain are a 0 = 4.189Å and c 0 = 5.394Å, which are close to those from experiment 43 .
To obtain the charge density, a spin-polarized selfconsistent calculation is performed with the relaxed structure for each strain. Using the obtained charge densities, the total energies E and E ⊥ are calculated in the presence of SOC where E and E ⊥ are the total energies with the Néel vector along Charge transfer between the Cr and Sb ions is analyzed by calculating the Bader charges. 44 The "net charge" on each atom is defined as the number of valence electrons for a given atom minus the Bader charge on the atom. The Bader charges are used to understand the effect of truncation of the bulk to a slab and the application of an electric field. Volumetric charge densities (ρ v ) at different electric fields are also calculated from the spin-polarized calculation. The difference between them is plotted to understand the charge transfer induced by the electric field over the slab.
To investigate the effect of electric field on the MAE of CrSb, a thin-film structure consisting of 2 unit cells along the c-axis (∼ 1.1 nm) plus a 15Å vacuum layer is considered. For the thin-film structure, an 11 × 11 × 1 kpoints grid with a 500 eV cutoff energy is used to ensure the same convergence criteria as the bulk structure. The structure is fully relaxed until all forces on each atom are less than 10 −3 eVÅ −1 without changing the volume. The dipolar correction is used for the relaxation process. The charge density is obtained by performing a spin-polarized self-consistent calculation without the electric field. Then the charge densities with different electric fields along caxis are obtained by relaxing the charges from the previous step. The MAE is calculated by the same method as for the bulk.  The effect of spin-orbit coupling (SOC) on the bandstructure is relatively large. A comparison of Fig. 2(a) to 2(c) shows that the SOC breaks the spin degeneracy throughout much of the Brillouin zone and creates anti-crossings at a number of band-crossing points. The largest band splitting occurs at Γ. The two bands that touch at 0.2 eV in the absence of SOC are split by ∼ 0.5 eV and the hole like band is pushed down below E F .
There is also significant electron transfer from the Cr atoms to the Sb atoms. The calculated Bader charges show a charge transfer of approximately 0.5 electrons from the Cr to the Sb atoms, and this transfer is identical for all Cr and Sb atoms in the bulk crystal. This corresponds to alternating sheet carrier concentrations of ±3.3 × 10 14 cm −2 on alternating Cr and Sb layers. Fig. 3(a) shows the MAE as a function of applied strain for bulk CrSb. The value at zero strain is E MAE = 1.2 meV/u.c. The positive sign of the MAE indicates that the Néel vector aligns along the c-axis (out-of-plane) independent of the strain. The monotonic increase in the MAE indicates that CrSb behaves like a magnet with a negative magnetostriction coefficient, since the tensile strain favors out-of-plane anisotropy. The magnetostriction coefficient (λ s ) is defined as where v, E, and ε are the Possion's ratio (0.288), Young's modulus (78.3 GPa), and strain, respectively 45 . The magnetoelastic anisotropy constant, K me , is calculated from the difference between two MAEs with and without strain (i.e., MAE(ε) − MAE(0)). The parameters v and E are taken from a previous study 46 . The calculated λ s for small strain (between −1 % to 1 %) is −19.8 ppm.
CrSb has a negative value of λ s , and the magnitude of λ s is similar to that of MnNi and MnPd 13 . For comparison with the thin film, we also define a strain coefficient as α ε = dE MAE /dε evaluated at ε = 0. The value for bulk CrSb is α ε = 0.13 meV/%strain. The response of the bulk MAE as a function of the electron number in the unit cell is shown in Fig. 4. In equilibrium, the unit cell has 22 valence electrons, which is denoted by the vertical line in the figure. The MAE decreases most rapidly when the CrSb is depleted, and it changes sign when the hole doping reaches 0.75/u.c. For electron depletion, the filling coefficient, defined as α n = dE MAE /dn, is 2.92 meV / electron / u.c. To understand the physical origin of the transition due to charge depletion, we consider the d-orbital resolved band structures for a Cr atom plotted for 2 different electron numbers in the unit cell, 22 (equilibrium) in Fig.  2(a) and 21 in Fig. 2(b). Within second-order perturbation theory, the MAE is approximately expressed as 47 where Ψ o (Ψ u ), E o (E u ), and ξ are the (un)occupied states, (un)occupied eigenvalues, and the spin-orbit coupling constant, respectively.L z andL x(y) are the out-of plane and in-plane components of the orbital angular momentum operator, and ↑ and ↓ denote spin-up and spin-down. The non-zero matrix elements in the Eq.
d xy |L y |d yz , and d x 2 −y 2 L y d xz . The largest contributions to Eq. (2) come from pairs of nearly degenerate occupied and unoccupied states near the Fermi level. In equilibrium (Fig. 2(a)), the main contributions to the MAE come from the spin-orbit coupling between occupied d ↑ xz and unoccupied d ↑ yz states at the K point and near the Γ point, and between occupied d ↓ xy and unoccupied d ↓ x 2 −y 2 states near the Γ point. All of these states couple through theL z operator, which results in the positive MAE value (out-of-plane anisotropy). When CrSb is depleted (see Fig. 2(b)), the entire band structure moves upward so that the main contributor states of the perpendicular anisotropy become unoccupied. This reduces the value of the MAE, and eventually reverses the sign for n = 21.25. We now consider a thin-film slab with a thickness of 2 unit cells or 1.1 nm as shown in Fig. 1(b). We note that the surface configuration shown is the most stable one. Inversion symmetry is broken, since one end is terminated with a Cr layer and the other end is terminated with a Sb layer. Thus, this thin film is a Janus structure. In the thin film, there is a net polarization of electron charge between the positively charged Cr layer on the bottom and the negatively charged Sb layer on the top. The excess charge on each atom in the bulk and in the slab is shown in Fig. 6(a,b), and it is clear that, in the slab, the charge transfer is no longer balanced layer-bylayer. The net charge polarization gives rise to a builtin electric field that alternates positively and negatively within the slab, but, its average value points from the positive Cr layer to the negative Sb layer parallel to the applied field shown in Fig. 1(b). Treating the Bader charges as sheet charges, the average electric field in the slab is 9.1 × 10 7 V/cm, and the average potential difference across the slab of 8.6 V.
The truncation of the bulk to a finite slab results not only in a loss of local balance between the positive and negative charges, but also in a global imbalance of the magnetic moments of the Cr ions. In other words, the cancellation of magnetic moments between alternating layers of Cr is no longer exact, and a small net magnetic moment exists in the slab. The magnetic moments on each Cr atom are listed in Table I with the numbering corresponding to that shown in Fig. 1(b). The magnitudes of the magnetic moments in the Cr layers monotonically decrease from bottom to top as the Cr atoms approach the Sb terminated end of the slab. The net magnetic moment of the slab is 0.8 µ B . The breaking of the spin degeneracy is readily apparent in the bandstructure of the slab shown in Fig. 7(a). The degeneracy between the up-spin and down-spin bands is broken, and the CrSb slab has become a ferrimagnet.   A third result of truncating the bulk to a slab is that the MAE decreases by a factor of 17 from 1.2 meV/u.c. to 0.07 meV/u.c., which corresponds to 0.15 erg/cm 2 . Furthermore, the sensitivity of the MAE to strain increases. The strain coefficient (α ε ) increases from 0.13 meV/%strain in the bulk to 0.47 meV/%strain in the 1.1 nm film, where the energies are per bulk unit cell (2 Cr atoms). The result is that a 1.6% uniaxial compressive strain along [1000] direction in the 1.1 nm thin film causes a 90 • rotation of the Néel vector from out-of-plane to in-plane as shown in Fig. 3(b).
A fourth result is that the MAE also becomes sensitive to an external electric field as shown in Fig. 5. In other words, the thin slab exhibits intrinsic VCMA. Typically VCMA is found when a magnetic layer is placed in contact with a heavy-metal layer that provides SOC. However, the Sb layers provide large SOC, and the terminating Sb layer serves as the HM layer, such that the CrSb slab has intrinsic VCMA. The MAE decreases linearly as the electric field is increased and changes sign at 1.4 V/nm, which demonstrates that the Néel vector is rotated 90 • from out-of-plane to in-plane purely by an electric field.
The sensitivity of the MAE to the applied electric field is given by the VCMA coefficient defined as where E ext is the external electric field and E I and I are the electric field and the relative dielectric constant in the insulator, repectively. In a typical experiment, the insulator would be an oxide layer. In our case, the insulator is the vacuum, so that I = 1 and E I = E ext . The value of β, evaluated at E ext = 0 V/nm, is −127.1 fJV −1 m −1 . This value of β is calculated for a freestanding CrSb slab, however, β is proportional to the dielctric constant of the insulating layer 16 , and it is also enhanced by inserting a heavy metal at the interface 23 .
Assuming that the insulator is MgO and using a relative dielectric constant of 7 at 750 GHz 48 , the MAE coefficient estimated from Eq. (3) becomes −889.7 fJV −1 m −1 . This magnitude of β is larger than that of a FeRh/MgO heterostructure 30 and comparable to those measured in heavy-metal/FM/MgO structures. 26 Also, compared to the G-type AFM FeRh/MgO heterostructure 30 , CrSb does not involve a magnetic phase transition for the spin reorientation.
To understand which Cr atoms contribute to the VCMA effect, we inspect the change of the charge and orbital magnetic moments of the Cr atoms in response to the applied field. The difference in the orbital magnetic moments (∆m l ) for each Cr atom in the slab at different electric fields is calculated and shown in the Table II. ∆m l is defined as ∆m l = m l,⊥ − m l, , where m l,⊥ and m l, are the orbital magnetic moments with the Néel vector being along [0001] and [1000] directions, respectively. The electric field changes ∆m l on Cr 1 by 3×10 −3 µ B and on Cr 4 by 1 × 10 −3 µ B . The interior atoms show negligible change. The change in charge with applied electric field, plotted in Fig. 6(c,d), shows that, among the Cr atoms, the only significant change in charge occurs on Cr 1 . This is to be expected, since the electric field is screened by the first and last layers from the interior of the CrSb metal. An external field of E ext = 3 V/nm induces a negative charge on Cr 1 of 0.018 electrons, as shown in Fig. 6(d). Both the change in orbital moment and the change in charge with applied electric field suggest that it is reasonable to focus on the Cr 1 atom for further discussion. To elucidate the physical origin of switching mechanism, the d-orbital resolved band structures for the Cr 1 atom under different electric fields are plotted in Fig.  7(b,c). At the electric field of −1 V/nm, the major contribution of the perpendicular anisotropy comes from the spin-orbit coupling between the unoccupied d ↑ xz and occupied d ↓ x 2 −y 2 states throughL y operator in the region 1. Although the states in the region 1 are coupled through the in-plane angular momentum operator, the contribution of the MAE is positive since they are from different spin channels. In the region 2, the occupied d ↑ xy states are coupled with unoccupied d ↑ xz states througĥ L x , which contributes the in-plane anisotropy. As the external field increases (see Fig. 7 (b)), the unoccupied d ↑ xz states in the region 1 move away from the Fermi level by 18 meV, which results in the reduction of the out-of-plane anisotropy since the denominator in the Eq. 2 increases. On the other hand, in the region 2, the occupied d ↑ xy states move closer to the Fermi energy by 67 meV as the electric field increases so that the in-plane anisotropy increases.

IV. SUMMARY AND CONCLUSIONS
The effects of strain, band filling, and electric field on the MAE of bulk and thin-film CrSb are determined and analysed. The magnitude of the bulk magnetostriction coefficient is comparable with those from other antiferromagnets and ferromagnets, however the MAE is large (1.2 meV/u.c.) and its sign cannot be changed by strain for bulk material. For bulk CrSb, depleting the electron density by 0.75 electrons per unit cell depletes the flat, nearly-degenerate d-orbital bands near the Fermi energy and causes a 90 • rotation of the Néel vector from outof-plane to in-plane. Truncation of the bulk crystal to a two unit-cell, 1.1 nm, thin film, breaks inversion symmetry, creates a large charge dipole and potential difference across the slab, and breaks spin degeneracy such that the CrSb slab becomes a ferrimagnet. The MAE is reduced by a factor of 17, from 1.2 meV/u.c. to 0.07 meV/u.c. (2 Cr atoms) or 0.15 erg/cm 2 , the strain coefficient is increased from 0.13 meV/%strain to 0.47 meV/%strain, and the sign of the MAE can be switched with 1.6% uniaxial compressive strain. The large SOC from the Sb combined with broken inversion symmetry of the thin film results in an intrinsic VCMA. With vacuum as the insulator, the calculated VCMA coefficient for the 1.1 nm thin film is −127.1 fJV −1 m −1 , and with MgO as the insulator, it is estimated to be −889.7 fJV −1 m −1 due to the increased relative dielectric constant. These values are comparable to those measured in other heavymetal/FM/MgO structures. 26