
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38483985
202316910
10.1073/pnas.2316910121
research-articleResearch ArticlephysPhysics426
Physical Sciences
Physics
Large nonlinear optical magnetoelectric response in a noncentrosymmetric magnetic Weyl semimetal
Shoriki Kentaro a 1 https://orcid.org/0000-0002-1321-0301

Moriishi Keigo a 1
Okamura Yoshihiro a 1 https://orcid.org/0000-0002-4987-7095

Yokoi Kohei b https://orcid.org/0000-0001-7871-5223

Usui Hidetomo c
Murakawa Hiroshi d
Sakai Hideaki d
Hanasaki Noriaki d https://orcid.org/0000-0002-4579-3302

Tokura Yoshinori a e f https://orcid.org/0000-0002-2732-4983

Takahashi Youtarou youtarou-takahashi@ap.t.u-tokyo.ac.jp
a e 2
aDepartment of Applied Physics and Quantum Phase Electronic Center, University of Tokyo, Tokyo 113-8656, Japan
bDepartment of Physics, Gakushuin University, Tokyo 171-8588, Japan
cDepartment of Applied Physics Shimane University, Matsue, Shimane 690-8504, Japan
dDepartment of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
eRIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
fTokyo College, University of Tokyo, Tokyo 113-8656, Japan
2To whom correspondence may be addressed. Email: youtarou-takahashi@ap.t.u-tokyo.ac.jp.
Edited by J. C. Davis, University of Oxford, Oxford, United Kingdom; received September 29, 2023; accepted February 12, 2024

1K.S., K.M., and Y.O. contributed equally to this work.

14 3 2024
19 3 2024
14 9 2024
121 12 e231691012129 9 2023
12 2 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Significance

ME coupling, cross correlation between magnetism and electricity, enables their mutual control and even causes unique optical functionality, which has so far been studied only partially in insulating multiferroics. Weyl semimetals provide an alternative platform for intriguing optical ME response through nontrivial band topology. We show the giant nonlinear optical ME response in noncentrosymmetric magnetic Weyl semimetal PrAlGe. It is found that the magnitude of SHG can be controlled by the magnetic field direction through interference between nonmagnetic and magnetic SHGs, which critically depends on the light-propagating direction. The observed magnetically induced nonlinear susceptibility is an order of magnitude larger than that of the prototypical ME material. These observations establish the emergence of strong nonlinear ME coupling in Weyl semimetals.

Weyl semimetals resulting from either inversion (P) or time-reversal (T) symmetry breaking have been revealed to show the record-breaking large optical response due to intense Berry curvature of Weyl-node pairs. Different classes of Weyl semimetals with both P and T symmetry breaking potentially exhibit optical magnetoelectric (ME) responses, which are essentially distinct from the previously observed optical responses in conventional Weyl semimetals, leading to the versatile functions such as directional dependence for light propagation and gyrotropic effects. However, such optical ME phenomena of (semi)metallic systems have remained elusive so far. Here, we show the large nonlinear optical ME response in noncentrosymmetric magnetic Weyl semimetal PrAlGe, in which the polar structural asymmetry and ferromagnetic ordering break P and T symmetry. We observe the giant second harmonic generation (SHG) arising from the P symmetry breaking in the paramagnetic phase, being comparable to the largest SHG response reported in Weyl semimetal TaAs. In the ferromagnetically ordered phase, it is found that interference between this nonmagnetic SHG and the magnetically induced SHG emerging due to both P and T symmetry breaking results in the magnetic field switching of SHG intensity. Furthermore, such an interference effect critically depends on the light-propagating direction. The corresponding magnetically induced nonlinear susceptibility is significantly larger than the prototypical ME material, manifesting the existence of the strong nonlinear dynamical ME coupling. The present findings establish the unique optical functionality of P- and T-symmetry broken ME topological semimetals.

Weyl semimetal
nonlinear optics
magnetoelectric effect
quantum geometry
multiferroics
JSPS KAKENHI 22H04470 Yoshihiro OkamuraYoshinori TokuraYoutarou Takahashi JST FOREST Program JPMJFR212X Yoshihiro OkamuraYoshinori TokuraYoutarou Takahashi JSPS KAKENHI 23H05431 Yoshihiro OkamuraYoshinori TokuraYoutarou Takahashi
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pmcSymmetry breaking is the most fundamental aspect to design the functional materials response in crystalline solids. A combination of inversion (P) and time-reversal (T) symmetry breaking is the typical example, which allows the mutual coupling between the dielectric and magnetic properties as exemplified by the magnetoelectric (ME) effect in multiferroic insulators (1–3). This enables the less-dissipative control of the magnetization and electric polarization using external fields and also generates the unique spin excitation endowed with the electric-dipole activity called electromagnon (4–6). Since the emergence of ME coupling is guaranteed by such symmetry breaking, the electromagnetic responses incorporating the ME coupling are expected even in the metallic systems. In this context, the unique transport phenomena of metallic systems without both P and T symmetry are attracting much attention (7–11). The notable example is nonreciprocal charge transport phenomena; the longitudinal resistance is changed by the reversal of each of electric current, polar axis, and magnetic field direction. This nonreciprocity has been observed in a variety of materials such as the Rashba semiconductor, noncentrosymmetric topological semimetal and superconductor-metal bilayer systems. These nonreciprocal responses often arise from the low-energy dynamics of conduction electrons strongly intertwined with both broken P and T symmetry. On the other hand, in the higher energy, i.e., optical regime, the ME response of metal is expected to be dominated by interband optical transitions, which should be essentially different from the DC charge transport phenomena. However, the optical ME response in P- and T-symmetry broken metals remains elusive, despite the significant potential for various intriguing photonic functionalities (7).

Nonlinear optical effect incorporating the dynamical ME coupling, i.e., magnetization-induced second harmonic generation (SHG) (12–17), which is suggested to be related to the quantum geometrical nature of electron (18–24), is one important target. The conventional nonmagnetic SHG is widely used for many photonic devices and for detecting the P symmetry breaking of crystalline solids. When the T symmetry is further broken, the additional magnetic SHG emerges, which can be controlled by the external magnetic field. The interference between nonmagnetic and magnetic SHGs gives rise to the functional optical responses such as nonreciprocal optical effect and optical polarization rotation (12–17). For Weyl semimetals hosting pairs of Weyl points with intense Berry curvature (25, 26), gigantic nonlinear optical response has been found in P-symmetry broken polar crystals (18, 19) and largely enhanced linear magneto-optical response arising from the band topology is demonstrated for T-symmetry broken magnets (27, 28). Therefore, a Weyl semimetal without both P and T symmetry provides a promising platform to explore the enhanced nonlinear optical ME effect.

Here, we demonstrate the large nonlinear optical ME response in the magnetic and polar Weyl semimetal PrAlGe. We clearly demonstrate the magnetic field switching and directional response of the SHG, both of which are attributed to the interference between nonmagnetic and magnetic SHGs arising from broken P and T symmetry of the bulk crystal. The nonmagnetic and magnetic nonlinear susceptibilities exhibit large values as compared to some prototypical materials.

Results

PrAlGe crystalizes into a body-centered tetragonal structure with space group I41md (Fig. 1A) (29–32). The stacking pattern of Al layers along the c axis breaks the P symmetry, resulting in the polar axis. Below TC ~ 15 K, the Pr spin moment is ferromagnetically ordered along the c axis with strong uniaxial anisotropy (Fig. 1 B and C). This material is also known to be a Weyl semimetal hosting more than 160 Weyl points within ±0.1 eV from Fermi level (20). While the nonmagnetic and magnetic PrAlGe are both Weyl semimetals due to the P symmetry breaking, where the magnetic order mainly causes Zeeman splitting, perturbatively modulating the electronic structure; this leads to further generation of Weyl points or their shift in momentum space (29) (see also SI Appendix). Some of the Weyl nodes near the Fermi level are clearly visualized by the ARPES measurement and the related large anomalous Hall effect has been discussed (29–32).

Fig. 1. Giant and anisotropic SHG in noncentrosymmetric magnetic Weyl semimetal PrAlGe. (A) Crystal structure of PrAlGe. The red arrows represent the Pr spin moments. (B) Temperature dependence of magnetization at a small B field (0.4 T) for B||c (red curve) and B||a (blue curve). (C) Phase diagram in magnetic field along the c axis as indicated by magnetization. The open circle represents the magnetic field where magnetic domains are fully aligned at each temperature. (D) The experimental setup of SHG. The 800-nm fundamental light is focused onto the ac-plane in the 45-degree oblique incidence. (E and F) The horizontally polarized (E) and vertically polarized (F) SHG intensity when rotating the incident polarization angle ϕ (D) at room temperature. The red and blue symbols represent the SHG response for PrAlGe and TaAs, respectively. The red and blue curves represent the fits based on the symmetry argument (see also SI Appendix).

We first characterize the SHG arising solely from the P symmetry breaking in the paramagnetic phase at room temperature. Fig. 1D shows a schematic of the experimental setup for SHG measurement; the 800-nm fundamental light pulse is focused onto the ac-plane sample in the 45-degree oblique incidence and the resultant SHG intensity is measured by the spectrometer equipped with the charge-coupled device (for more details, see Materials and Methods). Fig. 1 E and F, respectively, show the incident light-polarization dependence of horizontally polarized (Pout) and vertically polarized (Sout) SHG intensity at room temperature. We observe clear polarization anisotropy of SHG in both cases, in accord with the crystal symmetry. For example, the vertically polarized (Sout) nonlinear electric polarization, Px(2 ω), in the paramagnetic phase with point group 4mm is given as:[1] Px2ω=ε0χxzx2ωEzωExω,

where ε0, χijk(2ω), and Ej(ω) represent the permittivity of vacuum, second-order nonlinear optical susceptibility, and light electric field along j axis, respectively. The obtained pattern of SHG intensity, which is proportional to |Px(2ω)|2, well reproduces the observed anisotropy for light polarization of fundamental light (Fig. 1F, red curve). Likewise, we fit the horizontal Pout polarization and evaluate the relative magnitude of all the tensor elements; χzzz is the largest among three independent nonvanishing elements (for more details, see SI Appendix).

In comparison, we also measured the SHG response of a prototypical noncentrosymmetric Weyl semimetal TaAs with the same point group, which is known to show the record-breaking giant nonlinear susceptibility (18, 19); the 800 nm susceptibility of TaAs reaches ~10 times larger than the maximum response of the semiconductor GaAs. We found that two Weyl semimetals, TaAs and PrAlGe, show the comparable magnitudes of SHG response (Fig. 1 E and F), manifesting the giant nonlinear susceptibility for PrAlGe as discussed later in detail.

Having established the nonmagnetic SHG well above the TC, we pursue the magnetic SHG in the field-polarized ferromagnetic state below the TC. Below we focus on the vertically polarized (Sout) SHG response in the magnetic field (Fig. 2A). Above the TC, the SHG intensity is enhanced for the incident polarization angle ϕ of 45° and 135°, which shows the almost same magnitude and is independent of the magnetic field direction (Fig. 2B). In stark contrast, below the TC, the SHG shows different intensity between ϕ = 45° and 135° (Fig. 2C). The SHG intensity for ϕ = 135° is larger than that for ϕ = 45° in the positive field of +0.4 T (red symbols), which is reversed by the reversal of magnetic field; the SHG intensity for ϕ = 135° is smaller than that for ϕ = 45° (blue symbols). The field-induced switching magnitude, which is defined as Iϕ=135∘,+B-Iϕ=135∘,-B2[Iϕ=135∘,+B+Iϕ=135∘,-B]-Iϕ=45∘,+B-Iϕ=45∘,-B2[Iϕ=45∘,+B+Iϕ=45∘,-B], reaches almost 10 %; I(ϕ, B) represents the SHG intensity for each ϕ and magnetic field B.

Fig. 2. Magnetic field switching of SHG. (A) Experimental setup of magnetic field switching of SHG. The magnetic field is applied along the c axis and parallel to the polar axis. (B) The SHG response under the positive (red symbols) and negative (blue symbols) magnetic fields at room temperature, well above the TC. (C) The SHG response in the positive (red symbols) and negative (blue symbols) magnetic fields at 7.7 K, below the TC. (D) Temperature dependence of the magnetic-field-induced switching magnitude of SHG intensity (red symbols, see main text for definition) and of the longitudinal Kerr rotation angle θK (blue curve). (E) Simulation of SHG response in the magnetic field by assuming βχxyzmag/αχxzxnonmag = −0.05 (see SI Appendix for detail).

Temperature dependence of the switching magnitude is shown in Fig. 2D. The switching magnitude monotonically decreases as the temperature increases and totally disappears above the TC (Fig. 2D, red symbols). This behavior almost coincides with the temperature dependence of longitudinal Kerr rotation θK of incident fundamental light, which represents the magnitude of in-plane c-axis magnetization (Fig. 2D, blue curve; see also Materials and Methods). Therefore, these temperature dependences demonstrate that the observed magnetic field switching is caused by the spontaneous T symmetry breaking of the bulk crystal.

This magnetic field switching is well accounted for by incorporating the magnetic nonlinear susceptibility induced by the P and T symmetry breaking in addition to the nonmagnetic (crystallographic) SHG. Since the symmetry of the system turns into 4m’m’ below the TC, the nonlinear polarization Px(2ω) under the magnetic field is given by:[2] Px2ω∝αχxzxnonmagEzωExω+sgn(B)βχxyzmagEyωEzω,

where χxzxnonmag and χxyzmag, respectively, represent the nonmagnetic and magnetic nonlinear susceptibilities. α and β denote the prefactors including the Fresnel coefficient, beam conditions of incident light, and so on (SI Appendix). The first nonmagnetic term is present even above the TC, while the second magnetic term emerges only below the TC, whose sign is reversed by the reversal of magnetization, or equivalently magnetic field here, as represented by sgn(B) in Eq. 2. This magnetic SHG arises from the nonlinear ME coupling and can be observed through the interference with the nonmagnetic SHG (12). Since the SHG intensity proportional to |Px(2ω)|2 includes the cross term, or equivalently interference, between nonmagnetic and magnetic terms, the sign reversal of magnetic field results in the magnetic field switching of SHG intensity. To be more quantitative, we simulate the incident polarization dependence of the SHG intensity by assuming βχxyzmag/αχxzxnonmag = −0.05, as shown in Fig. 2E. The SHG shows different intensity between ϕ = 45° and ϕ = 135° at each magnetic field and their magnitude relation is reversed by the reversal of magnetic field, which well reproduces the experimental result including the switching magnitude.

To further explore the feature of magnetically induced SHG, we measure the SHG response by changing the propagating direction of the fundamental light; the fundamental light is irradiated from the left-hand side here (Fig. 3B), in contrast to the original setup (Fig. 3A). We find that the SHG response with respect to the magnetic field is totally reversed as compared to the original setup (Fig. 3 C and D); at the incident polarization angle ϕ = 45° (135°), the SHG intensity for the positive field is larger (smaller) than that for the negative field in the other setup (Fig. 3D), showing anticorrelation with the original setup in the magnetic field switching part (shaded areas in Fig. 3 C and D). This behavior is well explained in terms of the interference between nonmagnetic and magnetic SHGs. Since the light irradiation from the opposite side results in Ey→−Ey in Eq. 2, the nonlinear polarization Px(2ω) in this other setup is given by:[3] Px2ω∝αχxzxnonmagEzωExω-sgn(B)βχxyzmagEyωEzω.

Fig. 3. Directional response of SHG. (A and B) Two different experimental setups of magnetic field switching of SHG. The setup in A is the same as the original one shown in Fig. 2A. In B, the light is irradiated from the left-hand side, opposite to A. (C) The SHG response for the setup shown in A. (D) The SHG response for the setup shown in B.

Thus, the sign of the second magnetic term is reversed as compared to Eq. 2, which results in the sign reversal of the cross term in the SHG intensity when changing the light propagating direction, leading to the directional response of SHG.

We note that the nonlinear magnetization M(2ω) can contribute to the magnetic SHG in terms of the symmetry and plays the decisive role in the nonreciprocal optical diode effect (7, 16). However, the M(2ω) shows negligible response in the present material as suggested by the symmetry argument of directional SHG response (SI Appendix). The observed magnetic field switching and directional response mainly arise from the interference between nonmagnetic and magnetic SHGs but do not manifest the nonreciprocal response of the SHG light itself that requires the interference between P(2ω) and M(2ω) (7, 17).

Discussion

Finally, we compare the magnitude of nonmagnetic and magnetic nonlinear susceptibilities with other compounds, which is summarized in Table 1. The nonlinear susceptibilities of the present compound are large as compared to some noncentrosymmetric materials; in particular, the magnetic one of the present compound is ~30 times larger than that of the prototypical ME antiferromagnet Cr2O3 (12, 13). Although the magnetic SHG arising from the nonlinear ME coupling is generally much smaller than the nonmagnetic SHG, the magnetic SHG intensity in the present compound is strong enough to observe appreciable interference with the gigantic nonmagnetic SHG in the present material. This result suggests that the strong ME coupling emerges in optical regime for the present Weyl semimetal with P and T symmetry breaking. Recent theoretical works discuss the intimate relation of the second-order nonlinear optical effect including the magnetic term to the quantum geometrical nature of interband transition of electronic band structure (18–24), which is likely the case for the large magnetic and nonmagnetic SHG currently observed in PrAlGe. For example, the interband optical transition of type-I Weyl point located near ∑1-Z line in band calculation is included in the present photon energy range (29) (see also SI Appendix), which could strongly contribute to the SHG response through the quantum geometrical effect. This optical response for ME metal based on the band picture is essentially different from the previously reported optical ME responses in multiferroics that is explained by the optical transition among the energy levels of localized electron of each magnetic ion (12–17).

Table 1. Nonmagnetic and magnetic nonlinear susceptibility for various materials

Material	Nonmag/Mag	χijk	|χ| (pm V−1)	Fundamental wavelength (nm)	
PrAlGe	nonmag	χzzz	2,000	800	
TaAs		χzzz	7,200	800	
ZnTe		χxyz	500	800	
LiNbO3		χzzz	52	852	
PrAlGe	mag	χxyz	62	800	
Cr2O3		χxxy	2	1,160	
In the present experiment, we measure the SHG response for PrAlGe and TaAs. Other values are taken from the literature (13, 18).

The P- and T-symmetry broken Weyl semimetal can be viewed as the multiferroic (semi)metal and allows the emergence of ME coupling even in the presence of conduction electrons. We show that the magnetically induced SHG in the ferromagnetically ordered phase interferes with the nonmagnetic SHG, resulting in the magnetic field switching and directional response for light propagation. The observed magnetically induced SHG for PrAlGe is 30 times larger than that for classical ME antiferromagnet Cr2O3. These results clearly demonstrate the existence of strong ME coupling in the P- and T-symmetry broken Weyl semimetals as well as their versatile optical functionality. The concept of multiferroic metal coupled with the topological band structure paves the way for many ME phenomena appearing in linear and nonlinear electromagnetic responses, potentially leading to the exotic functionality of matter.

Materials and Methods

Single Crystal Growth.

Single crystals of PrAlGe were grown by the Al self-flux technique (32). Pr pieces, Ge, and Al powders with a molar ratio of 1:2:10 were put into the alumina under Ar gas atmosphere crucible and sealed in an evacuated quartz tube. They were heated to 1,150 °C and then slowly cooled to 750 °C. The single crystals were obtained after removing the excess Al flux by a centrifuge. Magnetization is measured by using Magnetic Property Measurement System (Quantum Design). Single crystals of TaAs were grown by a chemical vapor transport method using iodine as a transport agency (33). Ta foils and As, I2 pieces were put into a silica ampule and kept in the temperature gradient from 1,050 to 980 °C for 3 wk. The large single crystals with a diameter of 3 ~ 5 mm were obtained.

Second-harmonic Generation Measurement.

The optical experiments were performed in a variable-temperature optical cryostat with a permanent magnet. For a light source, we used a Ti:sapphire oscillator (MaiTai HP, Spectra Physics) with the pulse duration of 100 fs, the repetition rate of 80 MHz, and the center wavelength of 800 nm. The laser pulse is focused onto the ac-plane sample for PrAlGe and the (110)-plane for TaAs in the 45-degree oblique incidence and the reflected SHG is detected by using the spectrometer equipped with a liquid-nitrogen-cooled charge-coupled device. During the optical measurement below the TC, we apply the magnetic field of ±0.4 T to the sample, which almost fully polarizes the ferromagnetic domain (Fig. 1C).

Longitudinal Kerr Rotation Measurement.

The magneto-optical Kerr rotation of reflected fundamental light is measured by the balanced detection technique, which represents the magnitude of in-plane magnetization.

Supplementary Material

Appendix 01 (PDF)

We thank N. Ogawa for fruitful discussion. This work was partially supported by JSPS KAKENHI (Grant Nos. 22H04470, 23H05431) and JST FOREST Program (grant no. JPMJFR212X).

Author contributions

K.S., Y.O., and Y. Takahashi designed research; K.S., K.M., and Y.O. performed research; K.Y., H.U., H.M., H.S., N.H., and Y. Tokura contributed new reagents/analytic tools; K.S. and K.M. analyzed data; and Y.O. and Y. Takahashi wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix. Previously published data were used for this work (13, 18)

Supporting Information

This article is a PNAS Direct Submission.
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