Research ArticleCONDENSED MATTER PHYSICS

Proof of the elusive high-temperature incommensurate phase in CuO by spherical neutron polarimetry

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Science Advances  14 Feb 2020:
Vol. 6, no. 7, eaay7661
DOI: 10.1126/sciadv.aay7661
  • Fig. 1 Usual magnetic phase transition sequence in type II multiferroics.

    In the paramagnetic state (P), the magnetic moments are fluctuating (represented by blur) and condense into the high-temperature incommensurate phase AF3 at TN, which consists of an amplitude modulation with spins pointing along the magnetic easy axis. In the low-temperature incommensurate phase AF2, the spins adopt a cycloidal modulation before they align in a commensurate antiferromagnetic structure (AF1) at low temperature.

  • Fig. 2 Temperature dependence of the neutron intensity and polarization measured on the magnetic (000) + q peak.

    (A) The rocking scans over the incommensurate (ω ≈ −166.2°) and commensurate (ω ≈ −164.6°) peak positions at different temperatures (from 199.5 K shown in blue to 228.8 K shown in red with a linear color gradient in between) revealing the change of the propagation vector. The corresponding integrated intensities are depicted in (B). In (C), the Pyy term indicates the first-order transition from the AF1 to the AF2 phase at 213 K, which is also visible as a jump in the integrated intensities. At 228 K, a moderate increase toward a maximum value of 0.22 indicates the transition into the AF3 phase before the polarization drops to 0 above TN (a close-up of this region is shown in Fig. 5A). Note that the ordered moment is too small to see a change in the intensities. The black dashed lines mark the phase boundaries, and the red dotted line denotes the temperature at which the data collection within the AF3 phase was carried out.

  • Fig. 3 Results of the refinement to the SNP data.

    (A) The commensurate AF1 phase at 200 K, (B) the helical AF2 phase at 215 K, and (C) the so far uncharacterized AF3 at 228.8 K. Circles represent the observed polarization values, while squares show the refinement for the different magnetic Bragg reflections labeled on the horizontal axis. The color code identifying the different Pf,i elements is given as an inset in (C), e.g., a blue symbol with red edge means Pxz. All observed (obs) and calculated (cal) values are listed in table S1.

  • Fig. 4 Magnetic structure models.

    (A) Helical magnetic structure in the AF2 phase focusing on a part of a Cu-zigzag chain and showing a right-handed rotation (emphasized by the spiral) of the magnetic moments along the propagation vector. The circular envelope is depicted around the spins, where the b direction is represented in yellow and the α direction is in red. (B to D) Collinear sinusoidally modulated magnetic structures with magnetic moments along a direction in between b and α modulated by Γ1 + Γ2 (B), along the α direction modulated by Γ2 (C), and along the b axis modulated by Γ1 (D). The amplitude modulation is emphasized by sine waves.

  • Fig. 5 Simulated phase transition between the AF2 and AF3 phases.

    (A) The data points are the same as in Fig. 2B. Dashed and dotted lines represent the one-phase scenario for the two different possibilities: The SDW moment direction moves toward the b axis, or the SDW moment direction moves toward the α direction, respectively, with a simultaneous increase of the perpendicular component. Solid lines show the expected polarization values for a coexistence of two SDW structures in the AF3 phase with a gradual increase of the respective perpendicular component for decreasing temperature. (B) Perspective view on the spin envelopes at six different temperatures (shown in red and yellow for components along α and b, respectively) in comparison to the circular envelope deep within the AF2 phase (shown in gray). Rows 1 and 2 depict the simultaneous rotation and expansion (for the two cases mentioned above) of this envelope when going from the AF3 to the AF2 phase in the one-phase scenario. In the last row, the two-phase scenario is sketched, showing two envelopes at each temperature representing the two coexisting SDW phases. In each of those phases, the component perpendicular to the SDW moment direction increases upon cooling until the circular spin envelope is reached in both phases at T = 227.8 K.

  • Fig. 6 Comparison between SNP and conventional unpolarized neutron scattering.

    A simple Néel order with q = (0.5 0 0) on a primitive single-domain cubic lattice with one magnetic atom in the unit cell is assumed. The magnetic moment of fixed size (shown as blue arrows projected onto the a-c plane on the left-hand side sketches) is chosen (A) along the b axis, (B) 20° inclined from b toward c, and (C) 20° inclined from b toward the 〈101〉 direction. The sketches on the right-hand side show the corresponding local reference frames for a Q = (000) + q reflection together with the magnetic interaction vector M, the incident neutron polarization Pi (Pi||y in this illustration) and the final neutron polarization Pf. For each magnetic moment configuration, the modulus of the expected M (compared to the magnetic structure factor FM) and the two components of Pf are given. Note that unpolarized neutrons do not reveal any difference between cases (A) and (B) and that a 3% change of M in (C) can also be accounted for by a 3% smaller ordered moment. SNP clearly differentiates between the three cases—with a precision of approximately 1 to 2% of the respective components Pf,y and Pf,zindependently of the ordered moment size.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/7/eaay7661/DC1

    Section S1. Symmetry analysis in the extended little group (paramagnetic symmetry)

    Section S2. Order parameters in magnetic domains

    Section S3. Confirmation of the AF1 and AF2 phases

    Section S4. Incompatibility of the AF3 phase with a single irreducible representation

    Section S5. Theoretical explanation of the proposed AF3 phase

    Fig. S1. Order parameter in magnetic domains.

    Fig. S2. Temperature dependence of magnetic Bragg intensities.

    Fig. S3. Theoretical phase diagrams.

    Table S1. Transformation properties of the complex order parameters S1 and S2 in the extended little group (paramagnetic group).

    Table S2. Basis vectors Snx of the irreducible representations Γn.

    Table S3. Observed and calculated polarization matrix elements in the AF1, AF2, and AF3 phases.

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Symmetry analysis in the extended little group (paramagnetic symmetry)
    • Section S2. Order parameters in magnetic domains
    • Section S3. Confirmation of the AF1 and AF2 phases
    • Section S4. Incompatibility of the AF3 phase with a single irreducible representation
    • Section S5. Theoretical explanation of the proposed AF3 phase
    • Fig. S1. Order parameter in magnetic domains.
    • Fig. S2. Temperature dependence of magnetic Bragg intensities.
    • Fig. S3. Theoretical phase diagrams.
    • Table S1. Transformation properties of the complex order parameters S1 and S2 in the extended little group (paramagnetic group).
    • Table S2. Basis vectors Snx of the irreducible representations Γn.
    • Table S3. Observed and calculated polarization matrix elements in the AF1, AF2, and AF3 phases.

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