## Abstract

Nonlinear optical responses to external electromagnetic field, characterized by second- and higher-order susceptibilities, play crucial roles in nonlinear optics and optoelectronics. Here, we demonstrate the possibility to achieve ferroicity-driven nonlinear photocurrent switching in time-reversal invariant multiferroics. It is enabled by the second-order current response to electromagnetic field whose direction can be controlled by both internal ferroic orders and external light polarization. Second-order direct photocurrent consists of shift current and circular photocurrent under linearly and circularly polarized light irradiation, respectively. We elucidate the microscopic mechanism in a representative class of two-dimensional multiferroic materials using group theoretical analyses and first-principles theory. The complex interplay of symmetries, shift vector, and Berry curvature governs the fundamental properties and switching behavior of shift current and circular photocurrent. Ferroicity-driven nonlinear photocurrent switching will open avenues for realizing nonlinear optoelectronics, nonlinear multiferroics, etc., using the coupled ferroic orders and nonlinear responses of ferroic materials under external field.

## INTRODUCTION

Nonlinear light-matter interaction plays a key role in the understanding, probing, and ultimate control of light and matter at the nanoscale. In particular, materials with strong nonlinear optical responses are highly desirable for many scientific disciplines and technological applications (*1*–*3*), e.g., ultrafast nonlinear optics (*4*), nonlinear biosensing and imaging (*5*), efficient generation of entangled photon pairs for quantum computing and quantum sensing (*6*, *7*), and all-optical transistor and computer (*8*–*10*). Because of the odd parity of the two polar vectors, electric dipole and electric field, even-order nonlinear electrical susceptibility tensors vanish in centrosymmetric crystals (*11*, *12*), while odd-order responses are not limited by this constraint.

Among a plethora of optical responses, nonequilibrium direct current (DC) from periodically driven system under light field is of particular interest. One notable example is linear photovoltaic Hall effect, which has been predicted in graphene (*13*). In addition, there exist appealing second-order nonlinear DC responses to electromagnetic field in noncentrosymmetric solids, e.g., shift current (SC) and circular photocurrent (CC). Both were recently observed in Weyl semimetals (*14*–*18*). In parallel, recent experiments suggest that two-dimensional (2D) van der Waals layered materials exhibit symmetry-dependent strong nonlinear optical responses, such as second and third harmonic generation. It makes nonlinear optical spectroscopy a perfect facile tool for characterizing and quantifying 2D materials, e.g., elastic strain, crystal orientation, phase transition, crystal thickness, even and odd layer oscillations, etc. Strong nonlinear responses in 2D materials also suggest their great advantage for phase-matching free nonlinear optics (*19*–*21*).

SC, known as bulk photovoltaic current (*22*, *23*), refers to a photoexcitation of an electron from the valence to the conduction band with a simultaneous change in the phase of wave functions. Another type of nonlinear photocurrent, CC—also known as injection current, circular photogalvanic effect (*24*, *25*)—arises from the interference of wave functions upon photoexcitation associated with a phase difference between two linearly polarized light, which allows for phase-modulated nonlinear photocurrent with tunable magnitude and direction. For example, left and right circularly polarized light can induce opposite currents. Using semiclassical Boltzmann framework, Moore and Orenstein (*26*) and Sodemann and Fu (*27*) revealed the fundamental root of nonlinear currents in Berry curvature induced anomalous velocity of metallic materials. This framework includes intraband process as the product of Berry phase and the gradient of Fermi-Dirac distribution function, equivalent to Berry curvature dipole (*27*). The nonlinear response in this case considers the intraband process in metallic systems at the low-frequency regime. In intrinsic semiconductors or insulators, the gradient of the Fermi-Dirac distribution vanishes and thus the Drude-like SC and CC responses also vanish. However, nonlinear photogalvanic current persists in noncentrosymmetric semiconductors because of the nonlinear interband process, which wasn’t considered in the above intraband model.

Here, we demonstrate the possibility to achieve nonlinear photocurrent switching in time-reversal invariant multiferroics, namely, ferroicity-driven nonlinear photocurrent switching. More specifically, a Hall-like second-order direct photocurrent can be generated in time-reversal invariant systems upon electromagnetic field, whose direction can be facilely controlled by both internal ferroic orders and external light polarization and chirality. We provide a microscopic picture based on first-principles theory and group theoretical analysis of crystalline symmetry, time-reversal symmetry, permutation symmetry, gauge symmetry, and inherent causality. To illustrate the underlying mechanisms, we take monolayer group IV monochalcogenides [MX, with M = (Ge, Sn) and X = (S, Se, Te)], a ferroelectric-ferroelastic 2D multiferroics, as an example (*28*, *29*). Experimentally, Chang *et al*. (*28*) demonstrated that one of the MX compounds, atomic-thick SnTe, exhibits robust in-plane ferroelectricity at room temperature and that the application of voltage pulse between the scanning tunneling microscope tip and monolayer SnTe can manipulate the ferroelectric polarization through domain wall motion. Our previous theoretical study also demonstrated a wide range of kinetic barriers for coherent ferroic transition and domain wall energies (*29*). These experimental and theoretical studies suggested that it is possible to realize ferroelectric and ferroelastic switching in MX monolayers. In this work, using first-principles calculations and group theoretical analyses, we show that 2D MX monolayers exhibit large SC and CC responses that are dominated by topological quantities, i.e., shift vector and Berry curvature, respectively. Switching ferroelastic order flips the direction of nonlinear SC and CC photocurrent by ±90°, while switching ferroelectric order flips both nonlinear photocurrents by 180°. In addition, changing left and right circular polarizations of light will induce 180° flip in CC. The microscopic understanding of nonlinear photocurrent switching from first-principles theory, together with very recent discoveries of 2D ferroics and multiferroics and ferroelectric inorganic perovskites and hybrid organic-inorganic perovskites, will open a variety of new avenues for tunable and configurable nonlinear optoelectronics, bulk photovoltaics, and nonlinear multiferroics, etc.

## RESULTS

### Theory of SC and CC

Ferroicity-driven nonlinear photocurrent switching originates from the second-order photo-induced DC density ⟨*J*_{DC}⟩^{(2)}. Unlike linear photocurrent, the direction of ⟨*J*_{DC}⟩^{(2)} depends on intrinsic ferroic orders, which is the key to ferroicity-driven nonlinear photocurrent switching. ⟨*J*_{DC}⟩^{(2)} consists of two types of nonlinear photocurrents, SC and CC, which reflect the polarization change upon photoexcitation per unit volume*a*, *b*, and *c* are Cartesian indices. The electric field can be described using phasors ** E**(

*t*) =

**(ω) exp(−**

*E**i*ω

*t*) +

*c*.

*c*. For the linearly polarized light,

**(ω) is real, while for left and right circularly polarized light,**

*E***(ω) is complex,**

*E**E*(ω) = ∣

^{b}*E*(ω)∣

^{b}*e*

^{iφb}, with

*x*- and

*y*-polarized light, respectively.

The SC susceptibility tensor *25*) as*r** _{nm}* =

*i*⟨

*n*∣ ∂

_{k}∣

*m*⟩ and

*f*is the Fermi-Dirac distribution with

*f*≡

_{nm}*f*−

_{n}*f*, and

_{m}*m*and

*n*.

*d*dimension. The SC susceptibility tensor under the linearly polarized light can be rewritten in a more elegant expression

*(*

_{nm}**) is the phase factor of the Berry connection**

*k*The CC susceptibility tensor *25*) is given by*z**m* and *n*. The global one reads *bc*” indices in the CC susceptibility tensor are now absorbed in the index “*z*”.

SC and CC involve distinct physical processes. Figure 1 shows the corresponding microscopic picture using a two-band model. SC arises from the displacement of wave packet upon photoabsorption, while CC stems from the asymmetric transport of electrons and holes and the self-rotation of the wave packet. The latter induces orbital magnetic momentum coupled with the circularly polarized light. The intrinsic permutation symmetry of the electric field leads to

### Relation between SC and CC

Although SC and CC have different physical meaning, they are closely related. The derivative of CC susceptibility tensor

It shows that the two topological quantities, shift vector and Berry curvature, are closely connected. For a two-band model, *30*).

### First-principles calculation and group theoretical analysis of SC in 2D multiferroic MX

The symmetry property of linear susceptibility and nonlinear SC and CC susceptibility is governed by point group and permutation symmetry, which correspond to direct product Γ_{P} ⊗ Γ_{E}, Γ_{JSC} ⊗ Γ_{EE}, and Γ_{JCC} ⊗ Γ_{E×E*}, respectively. Here, we take monolayer group IV monochalcogenides [MX, with M = (Ge, Sn) and X = (Se, S)] as an example, which is a ferroelectric-ferroelastic multiferroics (*29*). In 2D MX with *C*_{2v} point group (see Fig. 2, A and B, for crystal and electronic structure, respectively) and its character table (table S1), we have Γ_{P} ⊗ Γ_{E} = 3*A*_{1} + 2*A*_{2} + 2*B*_{1} + 2*B*_{2}. Notice that the coordinate system here is different from the conventional one in table S1. Hence, there are three independent nonzero components in linear susceptibilities. The permutation symmetry further separates them into symmetric and asymmetric representations (*31*), Γ_{P} ⊗ Γ_{E} = Γ* ^{s}* + Γ

*, where Γ*

^{a}*= 3*

^{s}*A*

_{1}+

*A*

_{2}+

*B*

_{1}+

*B*

_{2}and Γ

*=*

^{a}*A*

_{2}+

*B*

_{1}+

*B*

_{2}. Moreover, because polarization

**, current**

*P***, and electric field**

*J***are all polar vectors, they share the same representation; thus, Γ**

*E*_{JSC}⊗ Γ

_{EE}= 7

*A*

_{1}+ 6

*A*

_{2}+ 7

*B*

_{1}+ 7

*B*

_{2}and

*x*- and

*y*-polarized light are considered. The corresponding SC along

*y*direction reads

*21*,

*32*). Furthermore, as shown in Fig. 2 (C and D), spin-orbit coupling (SOC) only slightly affects the SC in the case of GeS because of the weak spin-orbit interaction strength of Ge and S atoms.

*k*points around the Brillouin zone center (Γ point). The second peak at 2.8 eV denoted by green circle is apparently very strong. It comes from the transition at the

*k*points around the Brillouin zone boundary (

*X*point) in a butterfly shape as shown in Fig. 2C. However, SOC can have notable impact in other cases such as well-known 1

*H*-MoSe

_{2}and WSe

_{2}, and the results for all MX and MX

_{2}without and with SOC are shown in figs. S1 and S2, respectively.

The reciprocal vector-dependent contributions to SC are shown in Fig. 2 (E and F) for photon energy of 2.0 and 2.8 eV, respectively. They are determined by the product of SC susceptibility density, * _{cv}* − ω). The distribution of SC susceptibility density can be further understood by focusing on the frequency-independent terms determined by the product of dipole transition strength,

*k*points around the band gap, while

The quantitative relationship between shift vector and polarization difference was recently reported (*33*). In the presence of time-reversal symmetry, _{2} in the first Brillouin zone, demonstrating that its shift vector defined in Eq. 11 can go beyond its lattice parameter.

### First-principles calculation and group theoretical analysis of CC in 2D multiferroic MX

For the circularly polarized light along *z*, (*E × E*^{*})* _{z}* and axial axis

*R*share the

_{z}*B*

_{2}representation. Therefore, Γ

_{jx}⊗ Γ

*=*

_{Rz}*B*

_{2}⊗

*B*

_{2}=

*A*

_{1}, indicating that there is a nonzero CC response along

*x*direction

*z*direction, i.e., perpendicular to 2D plane. Figure 3 (A and B) shows two antisymmetric CC susceptibility tensor elements,

^{8}nm AV

^{−2}s

^{−1}, which allows us to estimate the generated nonlinear CC under continuous wave limit as follows. At room temperature, a typical relaxation time of the electrons in MX materials is around

*34*). μ

*is the mobility, and*

_{e}*m*

^{*}is the effective mass of electrons. Considering a regular laser pointer with an intensity of 1 mW/cm

^{2}and a 2D MX with an effective thickness of 1 nm, the induced CC

*J*

_{CC}can reach up to 10 to 30 μA/cm

^{2}, indicating that the current can even be observed using low-power continuous wave source (sheet photocurrent of

*35*). SC and CC are generated simultaneously under the circularly polarized light. It is possible to compare their peak amplitudes if we assume the same incident light intensity and a relaxation time of 10

^{−14}s for GeS. CC is larger than SC,

*J*

_{CC}/

*J*

_{SC}~5. The second-order nonlinear photocurrent response for different incident polarized light is summarized in table S2. Note that there is another nonzero element

*⊗ Γ*

_{jz}*=*

_{Rx}*B*

_{1}⊗

*B*

_{1}=

*A*

_{1}, suggesting that there exists a CC response along the

*z*direction when the incoming circularly polarized light is along the

*x*direction.

Figure 3 (C and D) shows the ** k**-resolved CC susceptibility in monolayer GeS under the circularly polarized light at two different frequencies (2.3 and 2.8 eV), demonstrating that the main response of the CC is localized around

*Y*point. Note that, for the same frequency of 2.8 eV, the SC (Fig. 2F) and CC (Fig. 3D) are very different from each other, as the SC is localized around the

*Y*point. The CC susceptibility is determined by the product of group velocity difference and Berry curvature

**and the axial vector**

*J***e × e***. Hence, it is nonzero for the point groups that allow optical activity or gyrotropy. Here,

**e**is the unit vector of light polarization. Figure 3E shows the group velocity difference between the highest valence band and the lowest conduction band, which confirms the time-reversal symmetry

**Δ**(

**) = −**

*k***Δ**( −

**) (see figs. S5 and S6 for the energy-dependent group velocity distribution). The Berry curvature**

*k**yz*plane) leads to opposite Berry curvature at (±

*k*,

^{x}*k*). The product of these two odd functions,

^{y}**Δ**(

**) and**

*k**C*

_{2v}point group. This is in direct contrast to 1

*H*-MoSe

_{2}, whose CC response vanishes because of its

*D*

_{3h}point group (see character table in table S1), as evident in its Berry curvature shown in fig. S7.

### Ferroicity-driven nonlinear photocurrent switching

The above group theoretical analyses and first-principles calculations illustrate the underlying selection rule and microscopic mechanism governing nonlinear SC and CC photocurrents. Because they are intimately related to the symmetry and topology, nonlinear SC and CC photocurrents are inherently coupled with the intrinsic ferroic orders in 2D multiferroics MX, giving rise to ferroicity-driven nonlinear photocurrent switching, which we will discuss below. Let us first inspect the coupling between ferroelectric order (*P _{y}*) and nonlinear SC and CC responses. Because both CC and SC are polar vectors, the sign of SC will flip upon ferroelectric polarization switch (

*P*→ −

_{y}*P*). Consequently, the sign of SC susceptibility tensor

_{y}

*r**(*

_{mn}**), shift vector**

*k*

*R**(*

_{mn}**), group velocity difference**

*k***Δ**

*(*

_{mn}**), and Berry curvature**

*k***Ω**

*(*

_{mn}**) as listed in Table 1.**

*k*Next, we examine the coupling between ferroelastic order (e.g., spontaneous strain ϵ* _{yy}* > 0 and ϵ

*< 0) and nonlinear photocurrent responses. Upon ferroelastic transition (ϵ*

_{xx}*→ ϵ*

_{yy}*and ϵ*

_{xx}*→ ϵ*

_{xx}*), shift vector*

_{yy}*xy*index. As a result, under the same linearly and circularly polarized light, nonlinear SC and CC photocurrent will change their direction by 90° upon ferroelastic transition, i.e.,

*D*

_{3h}, with a mirror plane perpendicular to

*x*axis, e.g., 1

*H*-MoSe

_{2}.

Because 2D MX has both ferroelectric and ferroelastic orders, it has four multiferroic (±*P*, ±ϵ) states, whose nonlinear photocurrent SC and CC are directly correlated as listed in Table 2. Here, we define ferroelastic strain +ϵ for ϵ* _{xx}* < 0 and ϵ

*> 0 and −ϵ for ϵ*

_{yy}*> 0 and ϵ*

_{xx}*< 0. Ferroelectric polarization*

_{yy}*P*could be ±

*P*if ϵ

_{x}*> 0 or ±*

_{xx}*P*if ϵ

_{y}*> 0. In contrast, linear optical susceptibility will not change with ferroelectric polarization switching because its matrix element*

_{yy}*P*, ϵ), nonlinear SC and CC current responses in 2D MX are bulk photocurrent response along different directions; thus, SC and CC can serve as a fundamental principle for real-space mapping of ferroelectric and ferroelastic orders.

## DISCUSSION

The above group theoretical analysis and first-principles calculations can be generally applied to other ferroic materials that can host ferroicity-driven nonlinear photocurrent switching. We can extend the above analysis to all 32 crystallographic point groups. There are 21 noncentrosymmetric point groups, 10 of which (*C*_{1}, *C _{s}*,

*C*,

_{n}*C*,

_{nv}*n*= 2, 3, 4, 6) have polar axis, a necessary but insufficient condition for ferroelectric materials. Assuming

*z*to be the polar axis,

*z*will transform as the total symmetric representation; therefore, Γ

*⊗ Γ*

_{jz}*⊗ Γ*

_{Ex}*and*

_{Ex}*Γ*⊗ Γ

_{jz}*⊗ Γ*

_{Ey}*always contain the total symmetric representation, indicating that nonvanishing SC is perpendicular to electric field and thus leads to a nonlinear transverse photocurrent. This transverse current cannot be observed from conventional linear Hall effect in time-reversal invariant materials. Furthermore, 18 of 21 noncentrosymmetric point groups including all the 10 polar classes are gyrotropic; hence, they all have nonvanishing CC.*

_{Ey}Although our focus here is on the ferroicity-driven nonlinear photocurrent switching based on interband process–induced second-order photocurrent, the similar effect can be also found in ferroelectric semimetals where static and low-frequency electric field–induced nonlinear Hall current can be switched upon ferroelectric switching. However, it should be emphasized that not all ferroic materials exhibit nonlinear photocurrent switching upon ferroic transition. Similar to the present case, it depends on the underlying symmetry transformation of ferroelectric transition. More excitingly, it can be readily extended to higher-order or other types of nonlinear effects, e.g., optically induced nonlinear magnetization switching and ferroicity-driven third-order nonlinear photocurrent switching. It will offer unprecedented opportunities for nonlinear optoelectronics, nonlinear optomagnetics, and nonlinear optoelectromagnetics, namely, nonlinear multiferroics, as potential routes to control polarization, charge current, magnetization, and spin current in an ultrafast, noninvasive manner without the stringent requirement of strongly coupled lattice, polarization, and magnetization in conventional multiferroics.

In summary, using group theoretical analyses and first-principles calculations, we have studied the microscopic mechanism of nonlinear photocurrent switching in time-reversal invariant multiferroics. Our results show that nonlinear photocurrent is highly sensitive to the symmetry of materials, including point group symmetry and time-reversal symmetry. This leads to ferroicity-driven nonlinear photocurrent switching—unique to multiferroics where the direction of second-order photocurrent is strongly correlated with intrinsic ferroic orders and external light polarization. The concept of ferroicity-driven nonlinear photocurrent switching illustrated here is not limited to 2D multiferroics, rather it can be generally applicable to many ferroics (e.g., ferroelectric inorganic perovskites and hybrid organic-inorganic perovskites) and even ferroelectric metals. One can envisage to directly control nonlinear photocurrent by switching ferroelastic strain and/or ferroelectric polarization accompanied by instantaneous direction and/or sign switching of the photocurrent. Furthermore, one may conduct high-resolution characterization of ferroelastic and ferroelectric orders and domain evolution using ultrafast optical techniques based on ferroicity-dependent nonlinear photocurrent. The present findings thus will open up avenues for realizing configurable nonlinear optoelectronics, bulk photovoltaics, nonlinear multiferroics, etc., using their ferroic orders and various nonlinear responses under external field.

## MATERIALS AND METHODS

### First-principles atomistic and electronic structure calculations

Ground-state crystal structures of MX were calculated by first-principles density functional theory (DFT) (*36*, *37*) implemented in the Vienna Ab initio Simulation Package (VASP) (*38*, *39*) with the projector-augmented wave method (*40*) and a plane-wave basis with an energy cutoff of 400 eV. We used the generalized gradient approximation of exchange-correlation functional in the Perdew-Burke-Ernzerhof (*41*) form and a Monkhorst-Pack *k*-point sampling of 12 × 12 × 1 for the Brillouin zone integration.

### First-principles nonlinear SC and CC calculations

To evaluate the nonlinear photocurrent responses, we developed a nonlinear optical code interfaced with first-principles DFT packages (e.g., VASP and Quantum-ESPRESSO). Second-order nonlinear photocurrent responses such as SC and CC are different from linear optical absorption, as the numerical integration of generalized Berry connection has to be performed over the whole Brillouin zone and multiple band indices for 27 third-rank tensor elements at each frequency. We therefore parallelized and benchmarked the code with tensor symmetrization and SOC taken into account. For the present calculations, we found that a dense 72 × 72 × 1 *k*-point sampling, 40 electronic bands without SOC (80 bands with SOC), and a total of 1000 frequency grids in an energy range of −6 to 6 eV are enough to achieve converged nonlinear photocurrent susceptibility tensors. The fundamental frequency ω in the denominator of susceptibility tensor carries a small imaginary smearing factor τ of 0.05 eV: ω → ω + *i*τ. We used the Sokhotski-Plemelj theorem to perform the integrals of Dirac delta function, which reads * _{mn}*. This always holds true in our case. Last, the single-particle approximation often underestimates the bandgap, and two-body and higher-order interactions may also affect the results. To check the effect of the single-particle approximation, we applied a scissor operator to correct both bandgap and matrix elements by using the optical bandgap obtained from quasiparticle GW and Bethe-Salpeter equation calculations. This approach is a reasonable remedy as the GW and DFT band structures are similar for the MX materials. Figure S8 shows the nonlinear photocurrent calculated with and without the scissor operator, and the main effect is the relative shift of the peak positions with slightly reduced peak intensity. This is expected as it mostly affects the denominator and delta function in the equations for SC and CC. Two-body and higher-order excitations in principle can be taken into account by incorporating many-body wave functions (e.g., exciton wave functions as a linear combination of electron hole pair excitations) into the SC and CC tensors, which may change the SC and CC peak position and amplitude. Nonetheless, the main conclusions on the ferroicity-driven nonlinear photocurrent switching behavior from the symmetry analysis and first-principles calculations shall still hold, and future efforts shall be made to include the exciton and other high-order excitation effect.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/8/eaav9743/DC1

Table S1. Character table for *C*_{2v} and *D*_{3h}.

Table S2. Second-order nonlinear photocurrent responses under different polarized light.

Fig. S1. SC and CC in monolayer MX (*C*_{2v}) and 1*H*-MX_{2} (*D*_{3h}) without SOC.

Fig. S2. SC and CC in monolayer MX (*C*_{2v}) and 1*H*-MX_{2} (*D*_{3h}) with SOC.

Fig. S3. Microscopic distribution and frequency-dependent shift photocurrent susceptibility in monolayer 1*H*-MoSe_{2} with *D*_{3h} point group.

Fig. S4. SC and its microscopic origin in monolayer 1*H*-MoSe_{2}.

Fig. S5. Group velocity *v*_{x} distribution in monolayer GeS.

Fig. S6. Group velocity *v*_{y} distribution in monolayer GeS.

Fig. S7. Berry curvature distribution in monolayer 1*H*-MoSe_{2}.

Fig. S8. SC susceptibility tensor elements (

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## REFERENCES AND NOTES

**Acknowledgments:**Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing.

**Funding:**This research was supported by the National Science Foundation (NSF) under award no. DMR-1753054.

**Author contributions:**X.Q. conceived the project and supervised the research. H.W. performed the calculations. H.W. and X.Q. developed the first-principles codes for computing second-order nonlinear SC and CC, analyzed the results, and wrote the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the corresponding author.

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