Spin ice Thin Film: Surface Ordering, Emergent Square ice, and Strain Effects

Motivated by recent realizations of ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ spin ice thin films, and more generally by the physics of confined gauge fields, we study a model spin ice thin film with surfaces perpendicular to the [001] cubic axis. The resulting open boundaries make half of the bonds on the interfaces inequivalent. By tuning the strength of these inequivalent “orphan” bonds, dipolar interactions induce a surface ordering equivalent to a two-dimensional crystallization of magnetic surface charges. This surface ordering may also be expected on the surfaces of bulk crystals. For ultrathin films made of one cubic unit cell, once the surfaces have ordered, a square ice phase is stabilized over a finite temperature window. The square ice degeneracy is lifted at lower temperature and the system orders in analogy with the well-known $F$ transition of the 6-vertex model. To conclude, we consider the addition of strain effects, a possible consequence of interface mismatches at the film-substrate interface. Our simulations qualitatively confirm that strain can lead to a smooth loss of Pauling entropy upon cooling, as observed in recent experiments on ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ films.

Figure 1

Spin ice thin film, thickness $L_h=1$ in the [001] direction. Open boundaries lead to orphan bonds (green) on the surfaces. Each spin amounts to a pair of $\oplus$ and $\ominus$ effective magnetic charges ($\pm q$) [8]. Negative ${\delta }_o$ favors magnetic surface charges $Q=\pm 2q$. Below the surface ordering temperature, at $T<T_{\mathrm{so}}$, the central tetrahedra (orange) develop into a square ice phase, as seen by the projection of the spins in plane. Shown, one of the ground states below the $F$ transition ($T<T_F$). $r_{nn}$ is the nearest-neighbor distance.

Figure 2

Specific heat $C_h$ for films (${\delta }_o=+4\mathrm{K}$ in orange and $-4\mathrm{K}$ in violet) of different thickness [$L_h=1$ (solid triangle), 2 (solid circle), 3 (square with dot inside)] and for bulk DSI (bulk DSI, black). Inset: low temperature $C_h$ for ${\delta }_o=-4\mathrm{K}$. Temperature axes: log (main), linear (inset). As the thickness of the film $L_h$ increases, the transition $T_F$ approaches the bulk $T_c\approx 0.18\mathrm{K}$ and becomes first order [44, 45].

Figure 3

Two-step ordering in thin films of thickness $L_h=1$ and ${\delta }_o=-4\mathrm{K}$: (a) specific heat and entropy, (b) order parameters and density $\rho$ of single charges (blue) inside the thin film, i.e., not on the surfaces. The latter is noticeably higher than for the bulk-DSI model (black). The surface ordering at $T_{\mathrm{so}}\approx 900\mathrm{mK}$ stabilizes a square ice model in the middle of the slab, as confirmed by the entropy in the intermediate region delimited by the dashed lines. Dipolar interactions ultimately lift the square-ice entropy at $T_F\approx 300\mathrm{mK}$ (orange) in favor of the same antiferromagnetic ground state as the $F$ model [39]. Temperature axes: log.

Figure 4

Strain effects smoothly lifts the Pauling entropy for $-0.2\mathrm{K}\lesssim {\delta }_d<0$ down to 0.4 K, while promoting bulk-DSI order for ${\delta }_d>0$ (e.g. ${\delta }_d=+0.2\mathrm{K}$). Here ${\delta }_o=0$. Thickness: $L_h=5$ (5 nm) as in the thinnest sample of Ref. [22].