Spin photogalvanic effect in two-dimensional collinear antiferromagnets

Spin photogalvanic effect (SPGE) is an efficient method to generate a spin current by photoexcitation in a contactless and ultra-fast way. In two-dimensional (2D) collinear antiferromagnetic (AFM) materials that preserve the combined time-reversal (T) and inversion (I) symmetry (i.e., TI symmetry), we find that the photogalvanic currents in two magnetic sublattices carry different kinds of spins and propagate in opposite direction if the spin-orbit coupling is negligible, resulting in a pure spin current without net charge current. Based on the first-principles calculations, we show that two experimentally synthesized 2D collinear AFM materials, monolayer MnPS$_3$ and bilayer CrCl$_3$, host the required symmetry and support sizable SPGE. The predicted SPGE in 2D collinear AFM materials makes them promising platforms for nano spintronics devices.

Except for the advantage of low dimensionality, 2D collinear AFM materials also exhibit ultrafast dynamic response, are robust against external magnetic perturbations, and have no stray field. [9][10][11] However, despite remarkable progress in exploring exotic phenomena in 2D collinear AFM materials, 2-6, 12 the applications of them so far have been restricted to introduce a ferromagnetic order or a net magnetization. The direct spintronic devices based on the intrinsic AFM order have yet to be determined.
One promising mechanism to directly use these 2D collinear AFM materials is the spin photogalvanic effect (SPGE), [13][14][15][16][17][18][19][20][21] which generates the spin current by photoexcitation in a contactless and ultra-fast way. In AFM materials, two sublattices with opposite spin polarizations are connected by certain symmetry. If the spin-orbit coupling (SOC) effect is negligible, the photogalvanic currents in two magnetic sublattices carry different kinds of spins and propagate in opposite directions, inducing a pure spin current in total. 21 The spin current here has a long spin diffusion length because spin is a conserved quantity. Moreover, the scattering induced by Joule heating and Oersted fields can also be well avoided.
In this letter, we predict that the 2D collinear AFM materials can be utilized to fabricate the nano spintronic devices via the SPGE. We study the SPGE in 2D collinear AFM materials, and demonstrate that the ̂̂ symmetry (the combination of time-reversal ̂ and inversion ̂ symmetries) guarantees the photogalvanic current with opposite directions and spin polarizations in two spin sublattices, and results in a pure spin current. As specific examples, we consider the Né el-type monolayer MnPS3 and the A-type bilayer CrCl3, and demonstrate, based on firstprinciples density-functional theory calculations (see Supporting Information), that these materials support a sizable SPGE.
Model and candidate materials. A collinear AFM material is composed of two FM sublattices with opposite magnetizations. These two sublattices are usually connected by a symmetry operation, leading to a zero-net magnetization. One of the most common symmetry operations to compensate for the magnetization is the ̂̂ symmetry, as shown in Figure 1a. In such an AFM material, even though both the time-reversal (̂) symmetry and the space inversion (̂) symmetry are broken, the combined ̂̂ symmetry is still preserved and connects the magnetic sublattices with opposite magnetizations. This symmetry enforces the energy bands of two sublattices are degenerate. In the absence of the SOC effect, the spin is a good quantum number, i.e. a conserved quantity. It allows us to investigate the SPGE of the ̂̂ preserved AFM materials in two sublattices with spin-up and spin-down magnetizations separately. 21 We therefore first describe the photogalvanic effect (PGE) in one spin sublattice. When an electron is pumped from the valence band to the conduction band by a linear polarized light, it generates a shift of wave packet in real space. This "shift vector" can be described by [22][23][24] where a is Cartesian index, is the phase factor of the interband Berry connection and is the intraband Berry connection matrix along a direction. The PGE element is determined by 21,22 where ( ) = ( ) − ( ) is the difference of occupation factors, ℏ ( ) = ( ) − ( ) is the difference of band energies, and is the frequency of light. Finally, the induced charge PGE current density J is where , , ∈ { , , }, and is the electric field of light along the direction. The shift vector is odd under ̂ symmetry, i.e. ( ) = − (− ). Therefore, the integration in Eq. (2) is enforced to be zero in a centrosymmetric material, leading to the vanishing of PGE. For the spin sublattices shown in Figure 1b and Figure 1c, the broken ̂ symmetry allows the PGE in each spin sublattice individually. Since the magnetic moments in each sublattice are in FM order, the PGE charge current is spin-polarized with the tensor ↑ ( ↓ ) for the sublattice with spin-up (spin-down) magnetization. The ̂ symmetry converts a spin-up electron to a spin-down one, and the ̂ symmetry guarantees the shift vector to be opposite, i.e., ,↑ ( ) = − ,↓ ( ) (see Figure   1b and Figure 1c). On the other hand, the optical transition possibilities (∝ ( ) ( )) of two spin states in the same k points are equal. As a consequence, in a ̂̂ preserved AFM material, the spin polarizations and directions of the PGE current in different sublattices are enforced to be opposite, i.e. ↑ = − ↓ . This results in a vanishing charge current, but a finite pure spin current, determined by = ↑ − ↓ , which leads to the SPGE in an AFM material with weak SOC effect. It is necessary to emphasize that the SPGE current here has longer spin diffusion length compared to the systems with strong SOC effect because of spin conservation.
Therefore, we can construct a spintronic device based on SPGE as shown in Figure 1d: a laser light beam irradiate to a 2D collinear AFM material with ̂̂ symmetry, and generates a pure spin current J s without a charge current J c , which can inject the spin current to the attached terminal in a contactless and ultra-fast way. There is a rich collection of layered magnetic materials. 25-27 However, not all these magnetic orders support the required symmetry. To determine the suitable AFM order for the SPGE, we use a honeycomb lattice model, a common magnetic structure that most 2D collinear AFM materials adopt. MnBi2Te4. 5,6 In those materials, the magnetic atoms are arranged in FM order within intralayer, and in AFM order between adjacent layers. Although ̂ symmetry is preserved for monolayer, it is enforced to be broken in bilayer due to the interlayer stacking, which makes the ̂̂ preserved  group, which contains a C3 rotation symmetry along the z-axis, and a C2 rotation axis around the y direction, as shown in Figure 3a. The ̂̂ symmetry that connects two sublattices are clearly reflected in the calculated spin density (Figure 3b). It turns out that the calculated spin density of the non-magnetic P and S atoms are identical for two spin states. Nevertheless, the spin density of the Mn atoms is distributed asymmetrically for the opposite spin states, which leads to the broken ̂ symmetry in two spin states. The density of the opposite spin sublattices can be transformed into each other by the application of the ̂̂ symmetry. Figure 3c shows the calculated band structure of monolayer MnPS3, which is consistent with the previous works. 33,40,41 A direct bandgap of 2.50 eV is found at the K points.  To proceed, we consider the SPGE in bilayer CrCl3. Bulk CrCl3 is an AFM semiconductor with TN=15.5 K. 47 It has been successfully exploited to the atomic limit recently, and the A-type AFM order is maintained down to bilayer. [48][49][50] In each monolayer, the Cr atoms form a honeycomb structure, and each Cr atom is surrounded by six Cl atoms in octahedral coordination.
An interlayer shifting appears when stacking the monolayers (Figure 4a). The magnetic moments of the Cr atoms are ferromagnetically coupled in the same layer while antiferromagnetically coupled between the adjacent layers, forming an A-type AFM order. Although the monolayer CrCl3 is centrosymmetric, the bilayer stacking breaks the ̂ symmetry. This broken ̂ symmetry is clearly shown in the calculated density for two different spin states, as shown in Figure 4b. ) of the PGE coefficients due to the C3 symmetry. We find that the PGE coefficients are overall smaller than those in monolayer MnPS3, and it is because that the broken ̂ symmetry originates from the adjacent layer. However, there is a large peak at ~4 eV for ↑(↓) ( ↑(↓) , ↑(↓) ) . The PGE coefficients of spin-down states are opposite to those of spin-up ones due to the ̂̂ symmetry, leading to the SPGE in bilayer CrCl3. A similar SPGE is also expected in even-layer CrCl3.
Whereas, for CrCl3 with odd layers, the ̂ symmetry is restored, which prevents the SPGE. This indicates an interesting odd-even layer character of the SPGE in few-layer CrCl3.  and also holds great potentials to realize long-distance spin transport. Furthermore, the polarizability of spin current is independent on the photon energy and polarization direction.
Finally, we note that the switching Né el vector is equivalent to exchange the positions of the two sublattices, leading to the reverse of SPGE coefficients (see Supporting Information). Therefore, our proposal also offers an efficient method to detect the Né el vector.

Conclusions.
In summary, we theoretically demonstrate the SPGE in 2D collinear AFM materials with weak SOC, which can be used to generate a pure spin current. Symmetry analyse shows that such SPGE is attributed to two copies of the PGEs with antiparallel polarized spin currents that propagate oppositely in two AFM sublattices. The first-principle calculations demonstrate that two experimental synthesized 2D collinear AFM materials: Né el-type monolayer MnPS3 and A-type bilayer CrCl3 satisfy the symmetry requirement and host

I. Magnetic structures of AFM materials
Most layer materials have stacking honeycomb structures. They can be divided into two broad families: transition metal halides (including both dihalides and trihalides) and transition metal chalcogenides. Their magnetic structures are summarized in Table S1, and their properties are summarized in the recent review papers. [1][2][3] According to the symmetry analysis in the main text, monolayer FM, zigzag-type, and stripe-type AFM materials cannot have the spin photogalvanic effect (SPGE) due to processing the inversion symmetry. In contrast, the Né eltype AFM materials and even-layer A-type AFM materials break the inversion symmetry, which are in red in Table S1. Except for the symmetry requirements, the materials should have weak SOC effects. According to the above requirements, we can choose suitable materials.

II. Calculation method
The first-principles calculations based on density functional theory (DFT) are performed by using the VASP package. General gradient approximation (GGA) according to the Perdew-Burke-Ernzerhof (PBE) functional are used. The energy cutoff of the plane wave basis is set to 400 eV. The Brillouin zones are sampled with a 12×12×1 mesh of -points. In order to simulate the monolayers, 20 Å vacuum layers are introduced. The vdW force with DFT-D2 correction is considered. Hubbards U term of 3 eV and 5 eV are added for Cr and Mn atoms to account for strong electronic correlations.
The Wannier function method has advantages over the DFT method to calculate the photogalvanic coefficient due to the high-efficiency in the dense k-mesh calculations and the unnecessary of a large number of unoccupied bands. 13 Therefore, the DFT Bloch wave functions are iteratively transformed into maximally localized Wannier functions by the Wannier90 code. 14,15 Mn-d and Se/P-p (Cr-d and Cl-p) orbitals are used to construct the Wannier functions for MnPS3 (CrCl3). The effective tight-binding Hamiltonian is obtained to construct the band structures. The PGE coefficients are calculated for spin-up and spin-down bands separately, and numerically evaluated by response theory. 13,16,17 Convergence test of k-mesh is preformed, and 500×500×1 k mesh is sufficient to calculate7 the photogalvanic effect (PGE) coefficients.
The 3D-like PGE coefficients are obtained assuming an active single-layer with the thickness of 13

= ,
where is the calculated PGE coefficient, and ( < ) is the thickness slab.  For 2D materials, , , ∈ { , } make sense. Therefore, the elements in red have no physical meaning in this case. Therefore, there is only one independent tensor element and = = − for monolayer MnPS3, which is consistent with the calculation results in the main text.

III. Symmetry of photogalvanic tensor
For bilayer and bulk MnPS3, the in-plane C3 rotation symmetry is broken, due to the interlayer stacking. Each sub-spin lattice can be described by C2 (2//y) symmetry, and the corresponding PGE tensor obeys where the , , and have no mathematical relationship anymore.
Each sub-spin lattice of bilayer A-type CrCl3 can be described by C3 symmetry, and its PGE tensor obeys

A. MnPS3
The calculated lattice constant and bandgap of MnPS3 are shown in Table S2, which show considerable consistent with the experiments. The band structures with SOC effects are shown in Fig. S1, which show weak SOC effects, indicating the SPGE model is suitable for MnPS3.  The calculated PGE coefficients of the bilayer and bulk MnPS3 is shown in Fig. S2, which is consistent with the above symmetry analysis. , , and have no mathematical relationship anymore because the C3 symmetry is broken in the bilayer and bulk MnPS3, yet show weak symmetry relation.

B. Bilayer CrCl3
The calculated lattice constants of the bilayer CrCl3 are shown in Table S3. The calculated band structures are shown in Fig. S3, which show weak SOC effects.

C. Switching the Magnetic order
The switching of magnetic order in MnPS3 and CrCl3 leads to the reverse of SPGE coefficients as shown in Fig. S4 and Fig. S5, respectively.