Research ArticleCONDENSED MATTER PHYSICS

Theoretical guidelines to create and tune electric skyrmion bubbles

See allHide authors and affiliations

Science Advances  15 Feb 2019:
Vol. 5, no. 2, eaau7023
DOI: 10.1126/sciadv.aau7023
  • Fig. 1 Concept to create ESBs.

    (A) Sketches of a typical Bloch-like MSB. (B) Unit cell of PTO, where arrows mark the displacements yielding a local polarization Pz or, equivalently, a vector field u||(0, 0, 1). Structure of the 180° FE DW of PTO at high (C) and low (D) T, as predicted in (23). (E) ND within a matrix of opposite polarization investigated in this work.

  • Fig. 2 Prediction of an ESB.

    Calculated polarization (A to C) and Pontryagin density (D to F) maps for our ND within a matrix in its ESB ground state (A and D), the same ND-ESB subject to an in-plane electric field along (1,1) (B and E, the field is indicated by a shadowed arrow) and the NDW-polar state stabilized for large enough field values (C and F). In (A) to (C), the color scale gives the out-of-plane Pz component, while the arrows correspond to the in-plane Px and Py. (G) Probability distribution for Q as a function of T. (H) Polarization as a function of in-plane electric field; black-filled squares give |Pz| as obtained at the middle of either matrix or ND (right vertical axis; the results for matrix and ND are essentially identical and very close to those for a monodomain state); blue-filled circles give the Px = Py components (left axis), obtained from a supercell average and normalized to the supercell volume; green open circles give the monodomain result for Px = Py. (I) Energy difference ΔE = ENDW-polarEND-ESB between the NDW-polar and ND-ESB states as obtained in a 16 × 16 × 1 supercell. Note that the ND-ESB and NDW-polar solutions are (meta)stable in the whole field range considered here. In (B) and (E), we show the ND-ESB solution at 500 kV/cm to better visualize the shift of the ESB center; in (C) and (F), we show the NDW-polar solution at zero field.

  • Fig. 3 Strain-driven transitions.

    (A) Polarization (top) and energy difference between the NDW-polar and ND-ESB states (bottom) as a function of the epitaxial constraint asub; details as in Fig. 2. (B) Position of the ESB center (top) and related susceptibility (bottom) as a function of asub (see the main text for definitions). In the top figure, the dashed line marks the regime where the polar NDW becomes the ground state and the polar ESB is a metastable solution. (C) Polarization map for the polar ESB state that appears for asub ≳ 3.95 Å. (D) For the same state, color map for the orientation of the in-plane polarization (Px, Py), evidencing the formation of 90° DWs.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/2/eaau7023/DC1

    Supplementary Discussion I: Terminology considerations

    Supplementary Discussion II: Minimal size of ESBs

    Fig. S1. Sketches of simulation supercell and skyrmion structures.

    Fig. S2. ESBs of different sizes and shapes.

    Fig. S3. Smallest ESB.

    Fig. S4. ESB as a four-state memory.

    References (3638)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Discussion I: Terminology considerations
    • Supplementary Discussion II: Minimal size of ESBs
    • Fig. S1. Sketches of simulation supercell and skyrmion structures.
    • Fig. S2. ESBs of different sizes and shapes.
    • Fig. S3. Smallest ESB.
    • Fig. S4. ESB as a four-state memory.
    • References (3638)

    Download PDF

    Files in this Data Supplement:

Navigate This Article