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Real-space imaging of phase transitions in bridged artificial kagome spin ice

Abstract

In frustrated spin systems, magnetic phase transitions underpin the formation of exotic, frustration-driven magnetic phases. Of great importance is the ability to manipulate these transitions to access specific phases, which in turn provides a means to discover and control novel phenomena. Artificial spin systems incorporating lithographically fabricated arrays of dipolar-coupled nanomagnets that enable real-space observation of the magnetic configurations provide such an opportunity. In particular, the kagome spin ice is predicted to exhibit two phase transitions, one of which is to a low-temperature phase whose long-range ground-state order has not been observed experimentally. To achieve this ordered state, we change the global symmetry of the artificial kagome system by selectively tuning the near-field nanomagnet interactions through nanoscale bridges at the lattice vertices. By precisely tuning the interactions, we are able to quantify the influence of frustration on the phase transition, finding that the driving force for spin and charge ordering depends on the degeneracy strength at the vertex.

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Fig. 1: Sample geometry with corresponding pairwise interaction strengths and vertex types.
Fig. 2: Magnetic configurations as a function of degeneracy strength.
Fig. 3: Relaxation to thermal equilibrium giving spin and charge order.
Fig. 4: Peak splitting in MSFs upon heating to the phase transition.
Fig. 5: Phase diagram of bridged artificial kagome spin ice.

Data availability

The data that support the findings in this study are available via the Zenodo repository https://doi.org/10.5281/zenodo.5550504 (ref. 36). Source data are provided with this paper.

Code availability

The MC code, written in Fortran, can be requested at khofhuis@gmail.com or peter.derlet@psi.ch.

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Acknowledgements

K.H., S.H.S., H.A. and L.J.H. acknowledge support from the Swiss National Science Foundation (project no. 200020_172774). Part of this project was performed at the Surface/Interface: Microscopy (SIM) beamline of the Swiss Light Source, Paul Scherrer Institute, Villigen PSI, Switzerland.

Author information

Authors and Affiliations

Authors

Contributions

K.H. and L.J.H. conceived the project. K.H. fabricated the samples, and designed and performed the experiments with support from H.A., Z.L., S.P. and A.K. K.H. analysed and interpreted the data with support from P.M.D. and L.J.H., and P.M.D. wrote the MC code used by K.H. to perform the correlator analysis and phase diagram determination with support from S.H.S. K.H. wrote the manuscript with input from all authors.

Corresponding authors

Correspondence to Kevin Hofhuis or Laura Jane Heyderman.

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Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Cristiano Nisoli, Susan Kempinger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–7 with captions. The source data for these can be found in the Zenodo repository.

Supplementary Video 1

XMCD image sequence of thermally activated nanomagnets relaxing to thermal equilibrium for the bridged artificial kagome spin ice with Ja/Jb = 0.5187 (corresponding to Fig. 3). The top left panel displays the XMCD images, obtained as a function of time at constant temperature T = 320 K (TB = 310 K), starting from an initial frozen as-grown out-of-equilibrium ice-rule-obeying phase, which relaxes towards the equilibrium LRO phase. The extracted spin configurations are given in the top right panel, from which the vertex charge ordering is extracted and indicated with yellow dots in the bottom left panel, while blue dots indicate disordered charges. In the bottom right panel, the corresponding formation of moment loops is displayed, indicated with magenta and green for the left- and right-handed chirality of the loops. The images are recorded with a field of view of 20 μm in the XPEEM.

Supplementary Video 2

XMCD image sequence of thermally activated nanomagnets with magnetic moments fluctuating at 310 K (TB = 310 K) for the bridged artificial kagome spin ice with Ja/Jb = 0.847 (Tcrit = 340 K). The top left panel displays the series of XMCD images obtained as a function of time at constant temperature after equilibrating for 20 min. The extracted spin configurations are given in the top right panel, from which the vertex charge ordering is extracted and indicated with yellow dots in the bottom left panel, while blue dots indicate disordered charges. In the bottom right panel, the corresponding formation of moment loops is displayed, indicated with magenta and green for the left- and right-handed chirality of the loops. The images are recorded with a field of view of 20 μm in the XPEEM.

Supplementary Video 3

XMCD image sequence of thermally activated nanomagnets with magnetic moments fluctuating at 337 K (TB = 310 K) for the bridged artificial kagome spin ice with Ja/Jb = 0.847 (Tcrit = 340 K). The top left panel displays the series of XMCD images obtained as a function of time at constant temperature after equilibrating for 20 min. The extracted spin configurations are given in the top right panel, from which the vertex charge ordering is extracted and indicated with yellow dots in the bottom left panel, while blue dots indicate disordered charges. In the bottom right panel, the corresponding formation of moment loops is displayed, indicated with magenta and green for the left- and right-handed chirality of the loops. The images are recorded with a field of view of 20 μm in the XPEEM.

Supplementary Video 4

XMCD image sequence of thermally activated nanomagnets with magnetic moments fluctuating at 339 K (TB = 310 K) for the bridged artificial kagome spin ice with Ja/Jb = 0.847 (Tcrit = 340 K). The top left panel displays the series of XMCD images obtained as a function of time at constant temperature after equilibrating for 20 min. The extracted spin configurations are given in the top right panel, from which the vertex charge ordering is extracted and indicated with yellow dots in the bottom left panel, while blue dots indicate disordered charges. In the bottom right panel, the corresponding formation of moment loops is displayed, indicated with magenta and green for the left- and right-handed chirality of the loops. The images are recorded with a field of view of 20 μm in the XPEEM.

Source data

Source Data Fig. 1

Folder Fig1b contains the data points of Fig. 1b in Excel format (Fig1b.xlsx), a high-resolution image of how the data looks in png and pdf format. Fig1a_SEM.png is the original SEM image, and Fig1c is attached as a PDF as file Fig1c.

Source Data Fig. 2

Folder with the 18 individual pictures in png format. The file names indicate their panel, and part e contains the bridge width from left to right [w0, w15,w17, w18, wb0].

Source Data Fig. 3

Folder with the original images for Fig. 3a panel 1 to 4 (left to right), then Fig. 3b panel 1 to 4 (left to right), and the data points of Fig. 3c,d in Excel format (Fig3cd.xlsx) with a corresponding image of how the data looks in png and pdf format.

Source Data Fig. 4

Original images of Fig. 4a for the temperatures 310, 337 and 339 K (left to right), and the data points of Fig. 4b in Excel format (Fig4b.xlsx) with a corresponding image in png and pdf format. Figure 4c for the temperatures of 260, 310, 340, 370 and 400 K (left to right) are provided as well in png format.

Source Data Fig. 5

The data points of Fig. 5 in Excel format (Fig5.xlsx) with corresponding png and pdf format of how the data looks.

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Hofhuis, K., Skjærvø, S.H., Parchenko, S. et al. Real-space imaging of phase transitions in bridged artificial kagome spin ice. Nat. Phys. 18, 699–705 (2022). https://doi.org/10.1038/s41567-022-01564-5

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