Crystal-chirality-dependent control of magnetic domains in a time-reversal-broken antiferromagnet

Chirality describes a lack of mirror symmetry and leads to functional properties such as natural optical activity and piezoelectricity. When time-reversal symmetry is also broken by magnetism, unconventional nonreciprocal phenomena emerge for photons, electrons, magnons, and phonons. Time-reversal symmetry can be broken even in antiferromagnets without net magnetization; recent theory shows such states are characterized by magnetic and toroidal multipole order parameters. Magnetic quadrupoles, which break both time-reversal and inversion symmetries, are central to linear magnetoelectric effects, nonreciprocal optics, and magnetotransport, and may be relevant to superconductivity. Combining quadrupoles with chirality could yield new phenomena, but these have been little explored. The authors propose that a Qxy-type magnetic quadrupole together with crystal chirality generates an orthogonal magnetization component relative to an applied magnetic field, whose sign reverses upon chirality switching. They test this in the chiral-lattice antiferromagnet Pb(TiO)6Cu4(PO4)4 (PbTCPO), which exhibits ferroic Qxy order, and show that this effect enables control of the quadrupole domain sign using only magnetic fields. Cluster mean-field calculations, cluster multipole decomposition, and a phenomenological Landau analysis support the mechanism and identify a key role of magnetic octupoles emergent due to chirality.

Prior work established nonreciprocal transport and optical effects in chiral ferromagnets and materials lacking inversion symmetry. Theoretical frameworks classify magnetic and toroidal multipoles under crystallographic point groups and relate them to response tensors. Linear magnetoelectric effects and toroidal moments have been extensively studied since early experiments by Astrov and subsequent developments in multiferroics. Magnetic quadrupoles have been implicated in magnetodielectric effects, nonreciprocal linear dichroism imaging, magnetopiezoelectric metals, and proposed as order parameters in cuprate pseudogap phases. Symmetry analyses distinguishing vector-like quantities under inversion, time-reversal, and mirror operations motivate coupling terms involving toroidal moments and chirality. For the square-cupola family A(TiO)Cu4(PO4)4, earlier experiments and theory established magnetoelectric quadrupole orders and exchange-striction-driven ME coupling, providing a platform to investigate chirality-multipole interactions.

Experimental: Single crystals of Pb(TiO)6Cu4(PO4)4 (PbTCPO) were grown by slow cooling of the melt. Crystal chirality was identified via optical rotatory sense using polarized light microscopy with incident light along Z and wavelength 500±10 nm; levo-rotatory crystals were defined as C− and dextro-rotatory as C+. Orientations were determined by Laue X-ray photographs. Monodomain C+ and C− crystals were used. Electric polarization Py was measured by integrating displacement current with an electrometer (Keithley 6517). For temperature dependence, samples were cooled from 10 K to 2 K with or without a poling field, poling removed, and current measured on warming. Ferroelectric hysteresis loops (Py vs Ey) were obtained by applying a triangle-wave voltage at 5 mHz using a voltage amplifier and function generator. A magnetic field H was applied in the XZ plane and tilted by Δθ from the X axis in the range −8° to +8°, controlled by rotating the sample about Y. Measurements probed Py under various μ0Hx (0–12 T) and small Hz arising from tilt to detect bias shifts indicative of Mz. The magnetic phase diagram with TN ≈ 7 K guided measurement points. The sign of quadrupole domains (Q±) was inferred from Py sign under Hx and bias response to Hz. Achieving single-domain states was attempted by cooling in tilted H without electric field. Theoretical: Cluster mean-field calculations employed an effective spin-1/2 model on Cu4O12 square cupolas with interactions H = Σ[J1 Si·Sj − Di·(Si×Sj)] + J2 Σ Si·Sj + ΣJ Si·Sj + ΣJ′ Si·Sj − gμB H·ΣSi. Parameters followed prior work: J1=1, J2=1/7, J=3/4, J′=1/100, |D|=1.1, with DM vectors tilted ~80° from Z toward X or Y. Chirality was introduced by a staggered in-plane rotation of DM vectors φDM = 5°, accompanied by a staggered rotation of the Cu4 squares (φ ≈ 5°) corresponding to the C+ crystal; the achiral reference had φDM = 0 respecting myz. Intercupola interactions were mean-field decoupled; intracupola interactions and Zeeman coupling were treated by exact diagonalization on four cupolas (two unit cells stacked along Z), yielding solutions equivalent to a single unit cell. Calculations evaluated mz vs hx (hz=0) for q± domains and free energy f vs hz at fixed hx to assess domain stability. Cluster multipole decomposition identified contributions to second-order nonlinear susceptibility χzxx, focusing on magnetic octupoles and toroidal quadrupoles. Phenomenological Landau analysis used symmetry of an achiral P4/nmm reference (point group 4/mmm) augmented by time reversal, then introduced chirality by removing myz, to derive invariant couplings linking Qxy, O-type magnetic octupole (O2), P, M, H, and E fields.

- Discovery of orthogonal magnetization Mz induced by the combination of magnetic quadrupole Qxy order and crystal chirality under an in-plane magnetic field Hx. This Mz is a second-order nonlinear response (∝ Hx^2) orthogonal to Hx and reverses sign with chirality and with switching of Qxy domains. - Material and symmetry context: PbTCPO crystallizes in chiral space group P42212 (enantiomorphic C±), featuring Cu4O12 square cupolas with staggered in-plane rotations (φ ≈ 4.5° at room temperature). The antiferromagnetic phase below TN ≈ 7.0 K has magnetic point group 4′22′, supports ferroic Qxy order and a linear magnetoelectric tensor with αyx = αxy ≠ 0. - Magnetoelectric characterization: Py emerges linearly with Hx (Py = αyx Hx), confirmed by hysteresis loops at 4.2 K for μ0Hx up to 12 T, showing bistable Q± domains with opposite Py at Ey=0. - Bias and domain selection by tilted magnetic fields: Small Hz produced by tilting H within the XZ plane breaks Q± degeneracy, shifting Py–Ey loops along Ey by ΔE. For C+ crystals at μ0H=9 T, ΔE > 0 when Hz>0 (Δθ>0), stabilizing the Q+ domain (Py<0, Mz>0), and ΔE<0 when Hz<0, stabilizing Q− (Py>0, Mz<0). The sign of ΔE reverses for C− crystals under identical tilt, evidencing that Mz reverses with chirality. ΔE is unchanged under simultaneous reversal of Hx and Hz, consistent with Mz ∝ Hx^2. - Electric-field-free domain control: Cooling through TN in tilted H (μ0H=9 T; Δθ=±8°) without any applied electric field yields single-domain states with Py ≈ +40 or −40 μC m−2 at 4.2 K (for Hz<0 or Hz>0 respectively), matching values inferred from hysteresis, demonstrating control of Qxy domain sign using magnetic field alone. - Theoretical validation: Cluster mean-field calculations with chiral spin-1/2 model (φDM=5°) produce finite mz of opposite signs for q+ and q− under hx with hz=0, whereas the achiral model (φDM=0) gives mz≈0. Calculated free energies vs hz at fixed hx show lifting of q± degeneracy with hz, selecting q+ for hz>0 and q− for hz<0, in agreement with experiment. - Multipole mechanism: Cluster multipole analysis and Landau symmetry arguments identify the dominant contribution to the second-order nonlinear susceptibility χzxx as a magnetic octupole O2. Chirality enables coexistence and coupling between Qxy and O2; switching Qxy flips O2, and vice versa. - Cross-control: Invariants imply that in the chiral case the sign of Qxy can be controlled by a purely magnetic nonlinear field combination HzHxHx, while O2 can be controlled by EyHx. The selected signs reverse upon chirality switching, explaining the observed chirality-dependent biasing. - Broader relevance: Among 122 magnetic point groups, six enantiomorphic antiferromagnetic groups (222, 422, 4′22′, 32, 622, 23) allow both linear ME effect and second-order nonlinear magnetization, suggesting applicability of the method beyond PbTCPO.

The work addresses whether crystal chirality can couple to antiferromagnetic multipoles to produce a controllable orthogonal magnetization in the absence of net magnetization. By combining Qxy order with chirality, the toroidal moment generated via the linear magnetoelectric effect under Hx couples to chirality to yield Mz ∝ C·Hx^2. Experiments on enantiopure PbTCPO directly probe this through Py–Ey hysteresis under tilted H: small Hz biases the quadrupole domains in a chirality-dependent manner, consistent with interaction −HzMz and reversal of Mz sign with chirality and Qxy domain. The independence of ΔE upon reversing H corroborates the second-order nature. Achieving single-domain states via magnetic field alone demonstrates practical control of antiferromagnetic ME domains without electric poling. Cluster mean-field theory quantitatively reproduces finite mz only in the chiral model and the hz-dependent domain selection, while multipole analysis connects the nonlinearity to a magnetic octupole O2 enabled by chirality. Thus, the findings resolve the proposed chirality–quadrupole coupling mechanism and reveal a general route to cross-control multipolar orders in antiferromagnets. The results are significant for antiferromagnetic spintronics, particularly in systems where electric-field control is hampered (e.g., metals), and suggest extending control via mechanical induction of chirality.

The study demonstrates that in the chiral-lattice antiferromagnet Pb(TiO)6Cu4(PO4)4, crystal chirality couples to magnetic quadrupole order to generate an orthogonal magnetization Mz under an in-plane magnetic field Hx. This Mz is a second-order nonlinear response whose sign reverses with chirality and with the Qxy domain. Exploiting this effect, the authors control the sign of Qxy domains using only a tilted magnetic field, achieving single-domain states without electric poling. Cluster mean-field calculations, cluster multipole decomposition, and phenomenological symmetry analysis attribute the effect to a chirality-enabled magnetic octupole O2 that couples to Qxy and mediates the nonlinear susceptibility χzxx. The work establishes cross-control between Qxy and O2 (via HzHxHx and EyHx, respectively) and points to broader applicability in enantiomorphic antiferromagnets. Future research could experimentally explore other chiral antiferromagnets (including metallic systems), implement current- or strain-based chirality induction, and quantify device-relevant response magnitudes and switching dynamics.

Explicit limitations are not extensively discussed. The experimental demonstrations focus on a single insulating compound (PbTCPO) at low temperatures and high magnetic fields, and generalization to other materials, especially metals or different magnetic point groups, remains to be validated experimentally.