Magnetic Proximity Evoked Colossal Bulk Photovoltaics in Crystalline Symmetric Layers

Bulk photovoltaic (BPV) effect, a second order nonlinear process that generates static current under light irradiation, requires centrosymmetric broken systems as its application platform. In order to realize measurable BPV photocurrent in spatially centrosymmetric materials, various schemes such as chemical doping, structural deformation, or electric bias have been developed. In the current work, we suggest that magnetic proximity effect via van der Waals interfacial interaction, a contact-free strategy, also breaks the centrosymmetry and generate large BPV photocurrents. Using the Bi2Te3 quintuple layer as an exemplary material, we show that magnetic proximity from MnBi2Te4 septuple layers yield finite and tunable shift and injection photocurrents. We apply group analysis and first-principles calculations to evaluate the layer-specific shift and injection current generations under linearly polarized light irradiation. We find that the magnetic injection photoconductivity that localized on the Bi2Te3 layer can reach over 70*108 A/(V2s), so that a 1D linear current density on the order of 0.1 mA/nm can be achieved under an intermediate intensity light. In addition to charge current, we also extend our discussions into spin BPV current, giving pure photo-generated spin current. The vertical propagation direction between the charge and spin photocurrents suggest that they can be used individually in a single material. Compared with previously reported methods, the magnetic proximity effect via van der Waals interface does not significantly alter the intrinsic feature of the centrosymmetric material (e.g., Bi2Te3), and its manipulation can be easily achieved by the proximate magnetic configurations (of MnBi2Te4), interlayer distance, and light polarization.

application platform. In order to realize measurable BPV photocurrent in spatially centrosymmetric materials, various schemes such as chemical doping, structural deformation, or electric bias have been developed. In the current work, we suggest that magnetic proximity effect via van der Waals interfacial interaction, a contact-free strategy, also breaks the centrosymmetry and generate large BPV photocurrents. Using the Bi2Te3 quintuple layer as an exemplary material, we show that magnetic proximity from MnBi2Te4 septuple layers yield finite and tunable shift and injection photocurrents.
We apply group analysis and first-principles calculations to evaluate the layer-specific shift and injection current generations under linearly polarized light irradiation. We find that the magnetic injection photoconductivity that localized on the Bi2Te3 layer can reach over 70×10 8 A/(V 2 ⋅s), so that a 1D linear current density on the order of 0.1 mA/nm can be achieved under an intermediate intensity light. In addition to charge current, we also extend our discussions into spin BPV current, giving pure photogenerated spin current. The vertical propagation direction between the charge and spin photocurrents suggest that they can be used individually in a single material. Compared with previously reported methods, the magnetic proximity effect via van der Waals interface does not significantly alter the intrinsic feature of the centrosymmetric material (e.g., Bi2Te3), and its manipulation can be easily achieved by the proximate magnetic configurations (of MnBi2Te4), interlayer distance, and light polarization.

Introduction.
Bulk photovoltaic (BPV) effect [1][2][3] that converts light irradiation into electric current in a single material has been attracting tremendous interests recently, which could avoid the traditional p−n semiconductor heterojunction fabrications [4,5]. In centrosymmetric ( ) broken systems, light induces anharmonic motion of carriers (electrons in the conduction bands and holes in the valence bands). Hence, two typical second order nonlinear optical processes, namely, sum frequency and difference frequency generations occur simultaneously [6,7]. The BPV effect arises from the difference frequency generation process under monochromatic irradiation (with angular frequency of ). Two major mechanisms of BPV processes that have been extensively studied are shift current and injection current generations, both of which originate from the geometric phases of electronic states (such as Berry phase) that strongly rely on symmetry constraints [8][9][10] The injection current density grows with time and saturates at the carrier relaxation time . Therefore, one usually discusses the photoconductivity , (0 bilayers with -symmetry [11]. Similar BPV currents have also been evaluated in the AF CrI3 bilayers [12,13] and their fundamental mechanisms have been revealed [14]. In these cases, the symmetry arguments and numerical calculations demonstrate that both NSC and NIC would vanish. For the -systems, generally speaking the linearly and circularly polarized light could induce MIC and MSC, respectively. Depending on specific materials, their existence can be determined by group theory (see Ref. [11][12][13][14] and Supplemental Material [15], SM for detail discussions).
In order to conceive finite photocurrents, as stated previously, one has to resort to -broken systems. This excludes a large number of spatially centrosymmetric materials, unless they are chemically doped (or alloyed) [16], structurally deformed [17][18][19][20], or electrically biased [21,22]. These schemes could effectively break in either transient

Methods.
Density Functional Theory. The first-principles calculations are implemented in the Vienna ab initio simulation package (VASP) [35,36] which adopts projector augmented-wave (PAW) method [37] and planewave basis set (cutoff energy of 350 eV) to treat the core and valence electrons, respectively. The exchange-correlation functional is using the generalized gradient approximation (GGA) in the solid state Perdew-Burke-Ernzerhof (PBEsol) form [38]. The periodic boundary condition is used and the first Brillouin zone is represented by a Γ-centered k-mesh with (15×15×1) grid [39]. In order to incorporate the strong correlations on magnetic Mn 3d orbitals, we add an effective Hubbard U (= 5.34 eV) correction according to the Dudarev scheme [40,41]. This value is widely used in previous works and has been demonstrated to yield results consistent with experimental observations [42][43][44]. One notes that other U values could give qualitatively same results. A vacuum space of at least 15 Å is added in the out-of-plane z direction to eliminate the artificial interaction between different images under periodic boundary condition. Self-consistent spin-orbit coupling interactions have been used in all calculations. The criteria of total energy and force component are set to be 1×10 −7 eV and 0.01 eV/Å, respectively. The vdW interactions are semiempirically included in the zero damping DFT-D3 method [45].
Bulk photovoltaic conductivities. The BPV photoconductivity coefficients are calculated using the Wannier representation, which includes Mn−d, Bi−p, and Te−p (covering all bands from −6.2 to 4.8 eV relative to the Fermi level) to fit the DFT electronic structures [46][47][48]. A k-mesh grid of 601×601×1 is adopted to integrate the optical conductivities, which is tested to achieve sufficient convergence accuracy.
According to the quadratic order Kubo perturbation theory, the BPV photoconductivity can be evaluated in various gauge schemes. The velocity gauge approach, which tends to diverge at low incident frequency, can be simplified into different photocurrents at the long carrier lifetime limit [49][50][51][52]. In detail, one can apply band theory to evaluate the NSC and MIC conductance coefficients separately, and Here the position operator matrix is defined as where = 〈 〉. Both of , (0, , − ) and , (0, , − ) feature electronic topological characters. One defines the shift vector under LPL irradiation between two bands as where is the complex phase of (= | | ). Therefore, it can be shown that , | | = Im ; which transforms as a polar vector. On the other hand, the quantum metric tensor is defined as which demonstrates the topological feature of MIC. Note that for -symmetric systems, the MIC vanishes (under LPL), while the NSC is symmetrically forbidden in systems.
As for spin photocurrent, we adopt the general definition of where is the spin operator (d is taken to be z in the current work).
Hence, in order to evaluate the spin current, we replace the velocity operator by . For example, the spin-velocity derivative can be written as [53] ; = ⟨ |{ , It would reduce to the charge current formula when one takes the spin operator into an identify matrix form. Since the MIC only incorporates the intraband contribution of spin current, we can include spin torque effect [19]. This is consistent with the "proper" definition of spin current via = ( ̂ ) [54]. In this case, the Onsager relation is maintained, so that the spin current obeys the conservation law of equation of motion.
In order to evaluate the layer-resolved current, one can introduce a layer projector operator, = ∑ | 〉〈 | ∈ with Wannier function | 〉 localized on a specific QL or SL [55]. Then we multiple this with velocity (or spin current) operator (or ), through which one could map the photocurrent onto each vdW layer. We have checked that the summation of each layer-resolved current on all the layers are almost the same as the total photocurrent. Hence, the interlayer photocurrent is marginal, as the vdW gap is ~3 Å. Note that in Eqs. (3) and (4), all the k-space integrals are taken in the 3D Brillouin zone, and the vacuum contributions have been included. We thus rescale these in-plane photoconductivity coefficients , (0, , − ) and , (0, , − ) by multiplying a factor , where is the simulation supercell lattice constant along z and is an effective thickness of the vdW layer [56,57]. In the current work, we use of MBT SL and BT QL to be 14 and 11 Å, respectively, which are taken from the thickness values when they form vdW bulk structures.

Results.
Photocurrents of MBT-MBT bilayer (Mn2Bi4Te8). We first examine the atomic geometries, electronic behaviors, and magnetic configurations of different MBT and BT superlattice slabs. In order to maintain the completeness of our study, we first explore the MBT bilayer (MBT-MBT) without BT intercalation [ Fig. 1(a)]. The Te atoms serve as termination layers between the MBT, leaving a vdW gap of 2.9 Å. Each Mn carries ~5 μB (Bohr magneton) magnetic moments pointing along the easy axis z, and the intralayer coupling is strongly locked to be FM. The interlayer MBT magnetic coupling prefers an AF configuration. This is usually terminated as the A-type AF state.
We reveal that the magnetic interlayer exchange energy Δ = − ( refers to total energy) is 0.43 meV per unit cell (0.04 μJ/cm 2 ). According to previous experimental facts, this AF ground state can be converted into FM via applying an external magnetic field below its Neel temperature (~12 T at 4.5 K or 6 T at 1.6 K) [58].   Figure S1 [15], which gives a larger direct bandgap. This is consistent with the results in previous works [59]. In the AF magnetic configuration ( ), each electronic band state is doubly degenerate without net spin polarization. The layer-dependent spin polarization exhibits a hidden feature. In our results, we use color scheme to illustrate spin polarizations 〈 〉 locally contributed from the SL1 and SL2, which shows opposite values in the two vdW SLs. Note that in our calculations, we have summed the doubly degenerated state contributions for each at k.
We next focus on the NSC and MIC photoconductivities. As discussed previously [60], symmetry operators pose constraints on both photocurrent direction and magnitude. Here we perform similar analysis on the two MBT SLs separately. The coordinates of each MBT SL contain a mirror reflection that is normal to x-direction, Fig. 1(a)]. Since Mn is magnetic along z, one has to multiple an additional timereversal operator, which gives ℳ . Thus, the local group on one MBT SL is reduced to be 3 ′, breaking the -symmetry. It then yields sizable NSC generation with one independent component (see SM) [15]. In detail, the ℳ maps BZ coordinate  Table I.  In order to verify these symmetry analyses, we plot our first-principles calculation results in Figs. 1(c)−1(e). Consistent with our argument, the , (0, , − ) is always zero, which is not shown for clarity reason. The , (0, , − ) on the SL1 reaches −90.9 μA/V 2 at an incident photon energy ω = 0.69 eV [ Fig. 1(c)]. Note that this value surpasses the NSC photoconductivity calculated in many other 2D materials, such as MoS2 [61], GeS [60], and α-NP [62]. In addition to charge current that uses electric charge as the degree of freedom to carry information, electronic spin can serve as another information carrier, which is facile to tune and easy to detect as well. In some recent works, photon illumination generated spin current has been predicted and evaluated in various material platforms [60,61]. Here we extend the LPL-induced charge photocurrent into spin photocurrent that remains further discussion or evaluation in -systems [63]. We adopt the anticommutation definition of spin current operator that has been widely used during the past decade. One has to note that spin torque effect may arise when spin operator is not a good quantum number under SOC [54,64]. This could be small in magnetic materials for intrinsic spin polarization. Under LPL, the spin current also composes MIC and NSC nature, which are depicted in Figure 3. One sees that the spin NSCs are We next show that magnetic configuration could tune these photocurrents. If the time-reversal AF configuration is used, the charge NSC remains their values, since they are -even. At the same time, the charge MIC reverses its direction, being a -odd quantity. However, for the spin current, since adds another -odd symmetry, the spin NSC remains its value while spin MIC becomes opposite. We plot our calculations in Fig. S2 [15]. The FM interlayer configuration, which has been successfully achieved in recent experiments, is equivalent to posing time-reversal onto one of these two SLs. Generally speaking, it reverses the charge MIC (and spin NSC) in such SL, while keeping charge NSC (and spin MIC). Note that the exact photoconductivity values are slightly changed due to band structure is altered (Fig. S3) [15]. In this regard, all the charge and spin currents become opposite in the two SLs, giving zero net (charge and spin) NSC and MIC. This is consistent with the fact that the FM MBT-MBT bilayer is -symmetric. These results are also plotted in Fig. S4 [15]. Therefore, we propose that one could apply magnetic field to toggle and achieve various layer-resolved charge and spin photocurrents.

Photocurrents of MBT-BT-MBT trilayer (Mn2Bi6Te11). We then intercalate a BT
QL inside the MBT-MBT bilayers, forming a MBT-BT-MBT trilayer [ Fig. 4(a)]. The two MBT SLs prefer a weak AF configuration, which is energetically lower than the interlayer FM state by 0.12 meV per unit cell. The significantly reduced magnetic exchange energy is ascribed by the long-range coupling between the two MBT SLs.
The band dispersion is plotted in Fig. 4 In order to further demonstrate the interfacial magnetic proximity effect, we artificially move the two MBT SLs away from the BT QL and calculate photocurrents.
The results are plotted in Figure S6 [15]. It shows that as the interlayer distance gradually increases from 2.9 Å (equilibrium) to 6 Å, the magnitude of MIC reduces but with evident values. When the MBT is sufficiently apart from BT QL, the MIC vanishes.
It clearly suggests that the interfacial magnetic proximity is the key mechanism to break in this situation, unlike previously reported routes.
We also plot the spin photocurrent of BT QL under x-LPL. According to symmetric argument, it has spin NSC nature and flows along y. Other spin BPV photocurrents are symmetrically forbidden. The results are shown in Figure 5. The perpendicular flow direction of charge and spin current indicates a pure spin current nature, similar as in other 2D nonmagnetic systems (such as MoS2 [61] and GeS [60]). We see that at ω = 1.36 eV, the , (0, , − )@QL reaches −59.7 μA/V 2 ⋅(ℏ/2e), surpassing that calculated values in the CrI3 bilayer [63] and BaFeO3 [67]. The distribution of ( , )@QL show clear mirror symmetry. We schematically depict the mechanism and propose the potential measurement setup in Fig. 7. As demonstrated previously, the charge and spin currents flow vertically [ Fig. 7(a)], which become silent after removing the AF sandwich layers, or when they form FM configuration. One notes that the AF configuration can be fine-tuned via external magnetic field, forming a double hysteresis loop on reversing the field sweep [ Fig. 7(b)]. Accordingly, the charge and spin BPV currents would emerge and diminish with respect to proximate magnetic orders [ Fig. 7(c)].
The photoconductivity that arises from the magnetic SL also provides a facile way to detect and measure axion insulating phase that usually appears in AFM layered materials. In axion insulators, the top and bottom layers that contribute 1/2 and 1/2 Chern numbers can be connected via a operator. The total Chern number is zero in an intrinsic axion insulator, which would provide a nonzero Chern number when is broken under external magnetic field. Here, we show that the charge MIC is even under . Hence, the total MIC is nonzero in an intrinsic axion insulator. This would provide an additional route for measuring axion insulators and does not require a magnetic field.
In our above evaluations, we only focused on the x-polarized LPL. Since the system is rotational invariant, the y-polarized LPL irradiation would induce similar (charge and spin) BPV photocurrents with a sign change. Hence, the carrier motion can be effectively manipulated by the polarization direction of LPL (obeying a sinusoidal function with respect to polarization angle). As for the circularly polarized light, it would introduce additional photocurrent via its phase factor. Generally, such phase factor results in 90°-rotated photocurrents. The shift and injection nature varies as well.
We summarize the mirror reflection, spatial central inversion, and time-reversal impacts on the LPL-induced charge and spin photocurrents in Table II One may wonder if out-of-plane BPV conductivity components of 2D materials can be similarly calculated and discussed. In the current theoretical framework, in-plane electric field component corresponds to normal incidence of light. Then the periodicity in the xy plane eliminates the edge/end effects. According to previous work [68], this is akin to the closed circuit boundary condition, where electric field E can be used as the natural variable. As is well-known, in finite sized system, the edge/end effect could accumulate induced charges and drastically change the boundary condition. The outof-plane electric field component corresponds to tangential incidence, which is experimentally challenging for thin films. In addition, the boundary condition becomes open circuit feature, and the charge accumulation on the film surface could be significant. In this situation, one should, in principle, adopt electric displacement D rather than E as the natural variable. On the other hand, the response function, formulated via Kubo perturbation theory [3], evaluates (charge and spin) current density. This definition is also in-plane, while it would be ill-defined for the out-ofplane current. Note that the z-direction is quantum confined for thin films, hence it is challenging to define a current along z. Recently, it is proposed that such out-of-plane response is actually electric dipole, and which requires evaluations of Wannier centers [69]. Nonetheless, the functions in the current work cannot be straightforwardly applied for out-of-plane component calculations.
We would like to remark that the magnetic direction would change the symmetry arguments since spin is a pseudovector. In this work, we only focus on the easy axis magnetic direction (z), which preserves the rotation. If the magnetic moment is aligned in the xy-plane, or when paramagnetic configuration emerges, the ℳ symmetry breaks, so that the photocurrents would change their mechanisms and flowing direction due to the reduction of its magnetic group. This is out of the scope of our current work, and will be discussed elsewhere.

Conclusion.
In the current work, we systematically calculate the layer-dependent photocurrent responses of MBT and BT superlattice thin films. Various sources of charge and spin BPV effects have been evaluated, and we provide symmetry constraints and arguments in detail. We show that even though a nonmagnetic BT QL is spatially centrosymmetric, one can break via applying magnetic proximity effect through the vdW interfaces.
In this regard, large charge and spin BPV currents can be generated. The direction of charge and spin current are vertical to each other, yielding pure charge and spin BPV photocurrent simultaneously. This strategy is different from previously proposedboken methods in otherwise nonmagnetic systems, such as introducing dopants, manipulating structures, or applying electric bias. The BPV charge and spin currents can be tuned by magnetic configuration, light polarization, and interfacial effects.