Peierls transition driven ferroelasticity in the two-dimensional $d\text{−}f$ hybrid magnets

For broad nanoscale applications, it is crucial to implement more functional properties, especially ferroic orders, into two-dimensional materials. Here, ${\mathrm{GdI}}_3$ is theoretically identified as a honeycomb antiferromagnet with a large $4f$ magnetic moment. The intercalation of metal atoms can dope electrons into Gd's $5d$ orbitals, which alters its magnetic state and leads to a Peierls transition. Due to strong electron-phonon coupling, the Peierls transition induces prominent ferroelasticity, making it a multiferroic system. The strain from unidirectional stretching can be self-relaxed via the resizing of triple ferroelastic domains, which can protect the magnet against mechanical breaking in flexible applications.

Figure 1

Physical properties of ${\mathrm{GdI}}_3$. (a) The structure of vdW layered ${\mathrm{GdI}}_3$. (b) The top view of a monolayer exfoliated from the vdW layered phase. The primitive cell is indicated by the dashed lines. Green: Gd. Purple: I. The spins of the Néel-type antiferromagnetic (AFM) ground state are indicated by arrows. (c) The band structure and density of states (DOS) of the ${\mathrm{GdI}}_3$ monolayer. The lowest conducting band is contributed by Gd's $5d$ orbitals, while the highest valence band is from I's $5p$ oribitals.

Figure 2

The Peierls transition of ${\left({\mathrm{GdI}}_3\right)}_2\mathrm{Li}$. (a) Structure of Li-implanted ${\mathrm{GdI}}_3$ monolayer. Blue: Li. Black rhombus: Primitive cell of FM or Néel AFM states. Red parallelogram: Primitive cell of the zigzag-AFM state. Blue rectangle: Primitive cell of the stripy-AFM state. (b) The Gd framework for the stripy-AFM phase. The structural dimmerization can be visualized clearly: the shorter $d_s$ between parallel-spin Gd-Gd pairs and longer $d_l$ between antiparallel-spin ones. (c) Fermi surface of the FM $P\overline{3}1m$ state, coming from Gd's $5d$ electrons. The hexagonal Fermi surface surrounding the $\mathrm{\Gamma }$ point provides the possibility for Fermi-surface nesting. (d) Phonon spectrum of the FM $P\overline{3}1m$ state, which indicates the dynamic instability. (e)–(h) Comparison of DOS near the Fermi level. (e) FM state on an optimized $P\overline{3}1m$ structure. (f) Stripy state on an optimized $C2/m$ structure. (g) Stripy state on a $P\overline{3}1m$ structure. (h) FM state on an optimized $C2/m$ structure.

Figure 3

(a) Schematic of electron distribution of Gd's $5d$ electrons. The electron spindles are along the $b$ axis in this $\alpha$ domain. The associated ferroelasticity can be characterized by the $b/a$ ratio of the black rectangle, which is 1.661, $\sim 4\%$ shorter than the original 1.732 for the $P\overline{3}1m$ phase. (b), (c) Two other ferroelastic domains with electron spindles along different directions, referred to as $\beta$ and $\gamma$ domains. (d) Origin of superelasticity. The macroscopic shape of the ${\left({\mathrm{GdI}}_3\right)}_2\mathrm{Li}$ monolayer can be deformed by external forces. The inner strain can be relaxed to some extent by the rotation of electron spindles, i.e., the resizing of the corresponding ferroelastic domains.

Figure 4

Results of ${\left({\mathrm{GdI}}_3\right)}_2\mathrm{Mg}$. (a) Electronic structure of the stripy-AFM state. The system remains a semiconductor, with two lowered $5d$ bands. Left: Band structure. Right: DOS. (b)–(d) Distribution of electrons. (b) and (c) are for two occupied $5d$ bands, respectively, and (d) are the total occupations.