Spin photogalvanic effect in two-dimensional collinear antiferromagnets

Recent discovered two-dimensional (2D) antiferromagnetic (AFM) van der Waals quantum materials have attracted increasing interest due to the emergent exotic physical phenomena. The spintronic properties utilizing the intrinsic AFM state in 2D antiferromagnets, however, have been rarely found. Here we show that the spin photogalvanic effect (SPGE), which has been predicted in three-dimensional (3D) antiferromagnets, can intrinsically emerge in 2D antiferromagnets for promising spintronic applications. Based on the symmetry analysis of possible AFM orders in the honeycomb lattice, we conclude suitable 2D AFM candidate materials for realizing the SPGE. We choose two experimentally synthesized 2D collinear AFM materials, monolayer MnPS3, and bilayer CrCl3, as representative materials to perform first-principles calculations, and find that they support sizable SPGE. The SPGE in collinear 2D AFM materials can be utilized to generate pure spin current in a contactless and ultra-fast way.


INTRODUCTION
Spintronics 1 , which exploits the spin degree of freedom of electrons in condensed matters for information storage and processing, has generated persistent interest in both physics and engineering. Conventional spintronics devices are usually constructed by threedimensional (3D) magnetic materials, and their application in nanoscale spintronics is hence strongly limited by the sizable thickness. The recently discovered two-dimensional (2D) van der Waals magnets naturally circumvent this limitation, thus bring an opportunity for next-generation nanoscale spintronic devices. In particular, 2D collinear antiferromagnetic (AFM) materials are especially interesting [such as bilayer CrX 3 (X = Cl, Br, I) 2-4 , evenlayer MnBi 2 Te 4 5,6 , monolayer MnPS 3 7,8 ]. Except for the advantage of low dimensionality, 2D collinear AFM materials also exhibit ultra-fast dynamic response, robustness 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][3][4][5][6]12 , the spintronic properties, such as magnetoresistance effect in CrI 3 4,12 and CrCl 3 13,14 due to the AFM-ferromagnetic transitions, and quantum anomalous effect Hall of MnBi 2 Te 4 6 induced by a finite magnetization, so far have been restricted to introduce ferromagnetic (FM) orders by strong magnetic fields. These require substantial electric currents and thus suffer from large energy dissipation. A spintronic property that utilizes the intrinsic AFM state will be more desirable for device application of the 2D collinear AFM materials.
One promising mechanism to use these 2D collinear AFM materials directly is the spin photogalvanic effect (SPGE) [15][16][17][18][19][20][21][22][23] , which generates spin current by photoexcitation in a contactless and ultra-fast way. In AFM materials, two sublattices with opposite spin polarizations are usually connected by certain symmetry. It was proposed that 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 23 . The spin current here has a long spin diffusion length because spin is a conserved quantity.
In this paper, we predict that 2D collinear AFM materials can be utilized to fabricate the nano spintronic devices via the SPGE. We focus on the 2D collinear antiferromagnets with time-reversalcombined-inversion ( b T b I) symmetry and weak SOC such as Néeltype monolayer MnPS 3 and A-type bilayer CrCl 3 , and demonstrate, based on first-principles calculations, that these materials support 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 are the b T b I symmetry, as shown in Fig. 1a. In such an AFM material, even though both the timereversal ( b T) symmetry and the space inversion ( b I) symmetry are broken, the combined b T b I symmetry is still preserved and connects the magnetic sublattices with opposite magnetizations. This symmetry enforces the energy bands of two sublattices to be 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 b T b I preserved AFM materials in two sublattices with spin-up and spin-down magnetizations separately using the model proposed by Young et al. 23 .
We first describe the photogalvanic effect (PGE) in one spin sublattice. When an electron is pumped from a valence band to a conduction band by a linearly polarized light, it generates a shift of wave packet in real space. This "shift vector" can be described by 24 where a is the Cartesian index, ϕ nm is the phase factor of the interband Berry connection and r a mm is the intraband Berry connection matrix along a direction. The PGE tensor element σ abc is determined by 23,24 σ abc ðωÞ ¼ À iπ e j j 3 2 h 2 R dk ½ P n;m f nm ðkÞR a nm ðkÞr b nm ðkÞr c mn ðkÞδ ω mn ðkÞ À ω ½ ; ( where f nm ðkÞ ¼ f n ðkÞ À f m ðkÞ is the difference of occupation factors, hω mn ðkÞ ¼ ϵ m ðkÞ À ϵ n ðkÞ is the difference of band energies, and ω is the frequency of light. The δ function originates from the quantum lifetime of electrons (see Supplementary Note 5). Finally, the induced charge PGE current density J is where a; b; c 2 fx; y; zg, and E a is the electric field of the linearly polarized light along the a direction. The shift vector is odd under b I symmetry, i.e. R a nm k ð Þ ¼ ÀR a nm Àk ð Þ. 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 Fig. 1b, c, the broken b I 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 σ " ðσ # Þ for the sublattice with spin-up (spin-down) magnetization. The b T symmetry converts a spin-up electron to a spin-down one, and the b I symmetry guarantees the shift vectors to be opposite, i.e., R a;" nm k ð Þ ¼ ÀR a;# nm k ð Þ (see Fig. 1b, c). On the other hand, the optical transition possibilities ð/ r b nm ðkÞr c mn ðkÞÞ of two spin states in the same k points are equal. As a consequence, in a b T b I preserved AFM material, the spin polarizations and directions of the PGE current in different sublattices are enforced to be opposite, i.e. σ " ¼ Àσ # . This results in a vanishing charge current, but a finite pure spin current, determined by σ S ¼ σ " À σ # , which leads to the SPGE in an AFM material with weak SOC effect. The model used here includes only the interband optical transition and is suitable for all the AFM semiconductors. For itinerant AFM materials, there should be a nonlinear Drude effect (intraband transition) arising from the Berry curvature dipole 28,29 . Since all the discovered 2D van der Waals antiferromagnets so far are semiconductors, the intraband optical transition of them in the SPGE is nonexistent and ignored in our work. Therefore, we can construct a spintronic device based on the SPGE as shown in Fig. 1d: A laser light beam irradiates to a 2D collinear AFM material with b T b I symmetry, and generates a pure spin current J s without any charge currents J c , which can inject the spin current to the attached terminal in a contactless and ultra-fast way. The spin current generated by this mechanism is expected to have a longer spin diffusion length compared to that generated by the mechanisms associated with SOC effect [15][16][17][18][19][20][21][22] . This is due to the spin conservation and the absence of momentum-dependent SOC field, which do not lead to the spin decoherence through the Dyakonov-Perel mechanism 30 during the scattering. The spin current may have interesting interactions with the local magnetic moments 31 , which is worth further study.
There is a rich collection of layered magnetic materials 32-34 . 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 that most 2D collinear AFM materials adopt. Figure 2 shows four common magnetic orders. The related materials are summarized in Supplementary Table 1. The zigzag-type and stripy-type AFM orders such as MPS 3 (M = Fe, Co, Ni) [35][36][37][38][39] and MSiSe 3 (M = V, Fe, Ni) 40 are shown in Fig. 2a, b, respectively. The b I symmetry of each sublattice can be clearly seen for these AFM orders, preventing the PGE in each sublattice and the SPGE of the AFM lattices. Figure 2c shows the Néel-type AFM order, where the adjacent magnetic moments are coupled with each other antiparallelly. This AFM order has been found in monolayer MnPS 3 7,8,41-43 and MnPSe 3 44 in experiments. In this AFM order, the b I symmetry of each sublattice is broken, and two sublattices are connected by the b T b I symmetry. This type of AFM order satisfies the symmetry for the SPGE. Different from the above AFM orders existing in the monolayer 2D materials, there is an A-type AFM order that can only emerge in multilayer 2D magnets (Fig. 2d), such as CrX 3 (X = Cl, Br, I) 2-4 and MnBi 2 Te 4 5,6 . In these materials, the magnetic atoms are arranged in the FM order within the intralayer, and in the AFM order between adjacent layers. Although b I symmetry is preserved for monolayer,  it is enforced to be broken in bilayer due to the interlayer stacking, which makes the A-type AFM materials preserve b T b I symmetry and compatible for the realization of the SPGE. Based on the above analyses, we can choose suitable 2D magnetic materials. As specific examples, we take Néel-type monolayer MnPS 3 and Atype bilayer CrCl 3 as two representative materials. The two experimentally synthesized 2D collinear AFM materials both have negligible SOC effects and suitable bandgaps for the SPGE.

First-principles calculation results
Transition-metal trichalcogenide MnPS 3 is an AFM material with a Néel temperature (T N ) of 78 K in bulk 45,46 . It has a layered structure, where the Mn atoms form a honeycomb lattice in each layer. A dumbbell-shaped P 2 S 6 unit is located at the center of each Mn hexagon (Fig. 3a). The magnetic moments of the nearest-neighbor Mn atoms are antiparallel to each other, forming a Néel-type AFM order. It has been successfully exfoliated down to single-layer recently 7,8,[41][42][43] , and the Néel-type AFM order can be maintained. The magnetic sublattices in monolayer MnPS 3 have the noncentrosymmetric structure with D 3 point group, which contains a C 3 rotation symmetry along the z-axis, and a C 2 rotation axis around the y direction, as shown in Fig. 3a. The b T b I symmetry that connects two sublattices is clearly reflected in the calculated spin density (Fig. 3b). It turns out that the calculated spin densities of the nonmagnetic P and S atoms are identical for two spin states. Nevertheless, the spin densities of the Mn atoms are distributed asymmetrically for the opposite spin states, which leads to the broken b I symmetries in two spin states. The densities of the opposite spin sublattices can be transformed into each other by the application of the b T b I symmetry. Figure 3c shows the calculated band structure of monolayer MnPS 3 , which is consistent with previous works 40,47,48 . A direct bandgap of 2.50 eV is found at the K points 47 . The bands contributed by spin-up and spin-down electrons are degenerate, and also show negligible SOC effect (see Supplementary Fig. 1). These characteristics support the SPGE in monolayer MnPS 3 .
There is only one independent tensor element ( σ "ð#Þ yyy ¼ Àσ "ð#Þ yxx ¼ Àσ "ð#Þ xxy ) of the PGE coefficients in each sublattice due to the D 3 symmetry (see Supplementary Note 2). The calculated PGE coefficients (Fig. 3d) are consistent with the symmetry. The PGE coefficients are zero at low photon energy, and become nonzero when the photon energy is larger than the bandgap. The value of the PGE coefficient can be as high as~10 μA=V 2 at 3.36 eV, which is comparable to the reported charge PGE in monolayer SnTe [49][50][51][52] and bilayer WTe 2 29 , and 3D conventional ferroelectrics (such as BiTiO 3 27 and PbTiO 3   26 ). We notice that the topological Weyl semimetals [53][54][55] , such as TaAs, have very large PGE coefficients due to the existence of Weyl cones 56 . Similarly, we speculate AFM topological semimetals possess large SPGE if symmetry is satisfied, which worthies further study.
In order to figure out the major contribution of this large PGE coefficient, we project the k-solved PGE coefficients of σ " yyy at 3.36 eV  Fig. 3c where the valence band shows flat character. In addition, the b T b I symmetry ensures the PGE coefficients for the spin-down state are opposite to those for the spin-up state. As a result, the total SPGE coefficient σ S ¼ σ " À σ # is expected to be sizable and can be easily detected under visible light energy through the magneto-optic Kerr effect or the inverse spin Hall effect. In addition, the SPGE is not limited to the monolayer, because the multilayer MnPS 3 also hosts the required symmetry (see Supplementary Note 4).
To proceed, we consider the SPGE in bilayer CrCl 3 . Bulk CrCl 3 is an AFM semiconductor with T N = 15.5 K 57 . It has been successfully exploited to the atomic limit recently, and A-type AFM order is maintained down to bilayer 13,14,58 . In each monolayer, the Cr atoms form a honeycomb structure, and six Cl atoms in octahedral coordination surround each Cr atom. An interlayer shifting appears when stacking the monolayers (Fig. 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 CrCl 3 is centrosymmetric, the bilayer stacking breaks the b I symmetry. This broken b I symmetry is clearly shown in the calculated densities for two different spin states, as shown in Fig. 4b. Figure 4c shows the calculated band structure of bilayer CrCl 3 . A bandgap of 2.61 eV is obtained. The Kramers degeneracy of the band structure results from the b T b I symmetry. The sizable PGE coefficients of each spin state are calculated and shown in Fig. 4d. There are two independent tensor elements (σ xxy ) of the PGE coefficients due to the C 3 symmetry. We find that the SPGE coefficients are overall smaller than those in monolayer MnPS 3 , because that the broken b I symmetry originates from the adjacent layer. However, there is a large peak around 4 eV for σ T b I symmetry, leading to the SPGE in bilayer CrCl 3 . A similar SPGE is also expected in even-layer CrCl 3 . Whereas, for CrCl 3 with odd layers, the b I symmetry is restored, which prevents the SPGE. This indicates an interesting oddeven layer character of the SPGE in the few-layer CrCl 3 .

DISCUSSION
According to the above calculations, the SPGE enables monolayer MnPS 3 and bilayer CrCl 3 to be promising candidates for nano spintronics devices, which can inject the pure spin current in a contactless and ultra-fast way. Furthermore, there are many other 2D collinear AFM materials that share similar lattice structures and satisfy the required symmetry of SPGE, such as Néel-type MnPSe 3 , A-type NiX 2 , FeX 2 (X = Cl, Br), CrBr 3 , etc. Nevertheless, the materials with strong SOC effects, such as the widely investigated A-type CrI 3 and MnBi 2 Te 4 , are not good candidates for the realization of the SPGE, though the symmetry requirements are satisfied. Bilayer A-type CrI 3 and MnBi 2 Te 4 have similar magnetic structures with bilayer CrCl 3 , but the spin conservation is heavily violated. However, the photoexcitation can still induce charge PGE in these materials, which have already been studied recently [59][60][61] . The SPGE proposed here takes advantage of the intrinsic character of the AFM state and holds great potentials to realize longdistance spin transport. Furthermore, the spin polarization of the spin current is parallel to the Néel vector, which offers the possibility to control the spin polarization of the generated spin current by manipulating the Néel vectors of these 2D antiferromagnets. Finally, we note that the switching Néel vector is equivalent to exchange the positions of the two sublattices, leading to the reverse of SPGE coefficients (see Supplementary  Figs 4 and 5). Therefore, SPGE also offers an efficient method to detect the Néel vector.
In summary, we theoretically demonstrate the SPGE in 2D collinear AFM materials, which utilizes the intrinsic AFM states to generate pure spin currents. Based on the symmetry analysis of possible AFM orders in the honeycomb lattice, we conclude the suitable 2D collinear AFM candidate materials for realizing the SPGE. Using first-principles calculations, we propose that Néeltype monolayer MnPS 3 and A-type bilayer CrCl 3 , two experimental synthesized 2D collinear AFM materials, satisfying the symmetry requirement and hosting detectable SPGE for promising spintronic applications. We hope our work can stimulate further experimental explorations and broaden the research scope of the 2D AFM spintronics.

First-principles calculations
The first-principles calculations based on density functional theory (DFT) are performed by using the VASP package. General gradient approximation according to the Perdew-Burke-Ernzerhof functional is 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 k-points. In order to simulate the monolayers, 20 Å vacuum layers are introduced. The vdW force with DFT-D2 correction is considered. Hubbards U terms of 3 eV and 5 eV are added for Cr and Mn atoms to account for strong electronic correlations. Wannier function method has advantages over the DFT method in calculating the photogalvanic coefficient due to the high-efficiency in the dense k-mesh calculations and the unnecessary of a large number of unoccupied bands 25 . Therefore, the DFT Bloch wave functions are iteratively transformed into maximally localized Wannier functions by the Wannier90 code 62,63 . Mn-d and Se/P-p (Cr-d and Cl-p) orbitals are used to construct the Wannier functions for MnPS 3 (CrCl 3 ). The effective tight-binding Hamiltonian is obtained to construct the band structures.

SPGE coefficient calculations
The PGE coefficients are calculated for spin-up and spin-down bands separately, and numerically evaluated by response theory as suggested in refs 24,25,51 . Convergence test of k-mesh is performed, and 500 × 500 × 1 k-mesh is sufficient to calculate the PGE coefficients. The 3D-like PGE coefficients are obtained assuming an active single-layer with a thickness of L active where σ slab is the calculated PGE coefficient, and L slab (L active < L slab ) is slab thickness.