Monte Carlo simulations of rare-earth holmium ultrathin films

Motivated by recent experimental results in ultrathin helimagnetic holmium films, we have performed an extensive classical Monte Carlo simulation of films of different thickness, assuming a Hamiltonian with six interlayer exchange constants. Both magnetic structure and critical properties have been analyzed. For $n>16$ ($n$ being the number of spin layers in the film) a correct bulk limit is reached, while for lower $n$ the film properties are clearly affected by the strong competition among the helical pitch and the surface effects, which involve the majority of the spin layers. In the thickness range $n=9-16$ three different magnetic phases emerge, with the high-temperature, disordered, paramagnetic phase and the low-temperature, long-range ordered one separated by an intriguing intermediate-temperature block phase, where outer ordered layers coexist with some inner disordered ones. The phase transition of these inner layers displays the signatures of a Kosterlitz-Thouless one. Finally, for $n\lesssim 7$ the film collapses once and for all to a quasicollinear order. A comparison of our Monte Carlo simulation outcomes to available experimental data is also proposed and further experimental investigations are suggested.

Figure 1

(Color online) (a) Film geometry: $N=n\times L_x\times L_y$ is the total number of spins, $n$ is the film thickness, i.e., the number of spin layer (free boundary conditions are taken along $z$ direction), and $L_{x,y}$ are the layer dimensions (periodic boundary conditions are applied along $x$ and $y$ directions). (b) Schematic representation of hcp Ho structure and exchange interactions: only $J_0$ (in-plane, green dashed lines), $J_1$ (NN planes, red dotted lines), $J_2$ (NNN planes, blue dot-dashed line), and $J_3$ (cyan dot-dot-dashed lines) are represented. The numerical values of the coupling constants employed in our simulations are $J_0=0.772\text{K}$, $J_1=0.346\text{K}$, $J_2=0.0754\text{K}$, $J_3=-0.0468\text{K}$, $J_4=-0.0638\text{K}$, $J_5=-0.00387\text{K}$, and $J_6=-0.0348\text{K}$.

Figure 2

Normalized magnetic vector profiles of each layer at $T=10\text{K}$, layer dimensions $L_{x,y}=80$, and different film thicknesses $n$: (a) $n=36$, (b) $n=16$, (c) $n=12$, (d) $n=11$, (e) $n=10$, (f) $n=9$, (g) $n=8$, (h) $n=7$, and (i) $n=6$. Error bars are included in point dimensions. Inset in (a): the magnified zone of the main graph (here shown for the sixth, 18th, and 30th planes from left to right along $m_l^x$, respectively) clearly shows that the helix pitch and lattice structure are incommensurate. Inset in (b): the magnification shows the magnetization of the first surface layers: first, 14th, second, 15th, third, and 16th from top to bottom along the $m_l^y$, respectively.

Figure 3

(Color online) Angle $\Delta {\varphi }_l={\varphi }_{l+1}-{\varphi }_l$ between magnetic moments in NN layers $\left(l+1,l\right)$ at $T=10\text{K}$ and $L=80$ for representative thicknesses: $n=16$ (black circle), $n=12$ (green diamond), $n=9$ (red square), and $n=6$ (blue up-triangle). Error bars included in the point dimensions.

Figure 4

(Color online) Structure factor $S\left(0,0,Q_z\right)$ (continuous line) vs $Q_z$ at $T=10\text{K}$ and layer dimensions $L_{x,y}=80$ for different film thicknesses: (a) $n=36$; (b) $n=16$; (c) $n=12$; (d) $n=11$; (e) $n=10$; (f) $n=9$; (g) $n=8$; (h) $n=7$; (i) $n=6$. Red dashed lines and green dot-dashed lines are the structure-factor components along the $x$ or $y$ spin space directions (see text). $Q_z$ is measured in reciprocal-lattice units $2\pi /c$ (Ref. 49).

Figure 5

(Color online) (a) Susceptibility and (b) binder cumulant vs temperature of the order parameter $M_l$ for thickness $n=12$ and $L=12$ (black circle), 16 (red square), 24 (green diamond), 32 (blue up-triangle), 40 (purple down-triangle), and 48 (brown star). $l$ runs from the first to the sixth spin layers. Close to the critical temperature $T_{C,12}\left(l\right)$, the continuum lines are obtained by multiple-histogram technique.

Figure 6

$T_{C,n}\left(l\right)$ vs layer index $l$ for (a) $n=20$, (b) 16, (c) 12, (d) 9, and (e) 8. Vertical dashed lines denote the position of the bisecting plane of the film, beyond which single-layer properties repeat by symmetry.

Figure 7

(Color online) Angle formed by the magnetization vectors of NN planes for $n=12$ and $L=48$ at different temperatures: $T=10\text{K}$ (solid black circle), $T=110\text{K}$ (open green diamond), and $T=118\text{K}$ (open red square) (Ref. 50). Error bars lie within point size.

Figure 8

(Color online) Temperature evolution of $S\left(0,0,Q_z\right)$ vs $Q_z$. (a) Thickness $n=12$ and $L=48$ at $T=10\text{K}$ (black line and circle), 113 K (red line and square), 116 K (blue line and triangle), and 121 K (green line and diamond). Dashed black line: $S\left(\vec{Q}\right)$ of the saturated block structure $\uparrow \uparrow \uparrow \uparrow \uparrow \circ \circ \downarrow \downarrow \downarrow \downarrow \downarrow$. Both the black curve $\left(T=10\text{K}\right)$ and the dashed one have been divided by a factor equal to 10. (b) Thickness $n=8$ and $L=48$ at $T=10\text{K}$ (black line and circle), 50 K (red line and square), 90 K (blue line and up triangle), 105 K (green line and diamond), and 110 K (orange line and down triangle). All temperatures are lower than $T_{C,8}\left(1\cdots 8\right)$.

Figure 9

Logarithm of the susceptibilities ${\chi }_{M_5}$ (square) and ${\chi }_M$ (circle) vs the logarithm of the lateral film dimensions $L$ (error bars included in the symbols), for $n=9$, at $T_{C,9}\left(5\right)$ and $T_{C,19}\left(1\cdots 4,6\cdots 9\right)$, respectively.

Figure 10

(Color online) $Q{\text{max}}_z$ vs temperature for some thicknesses: $n=16$ (black line and triangle), 9 (red line and circle), 8 (green line and square), and 7 (blue line and diamond). For $n=8$ the stability of the fan phase with respect to the collinear one is reached only below $T\approx 90\text{K}$ (see text).

Figure 11

Specific heat $C_v$ vs temperature for thickness $n=12$ and lateral dimensions $L=48$ (star), 32 (triangle), 24 (diamond), and 16 (square).

Figure 12

(a) Chirality vs. temperature at $n=12$. The normalization factor is fixed by the bulk factor $\text{sin}{Q}_z^{\text{bulk}}$ [see Eq. (6)]. Error bars are smaller than point size. The continuum lines in the inset are obtained by multiple-histogram algorithm. (b) ${\chi }_{\kappa }$ obtained by multiple-histogram algorithm. The largest relative error is 0.5% for $L=48$.

Figure 13

(Color online) (a) ${\partial }_{\beta }⟨\kappa ⟩$ vs temperature for $L=12$, 16, 24, 32, 40, 48 (symbols as in Fig. 5). (b) $T_N\left(12\right)$ plotted vs $L^{-1/\nu }$ obtained by finite-size scaling extrapolation of the three observables, with fitted $\nu$ value. Inset: plot of the maximum value of $\text{ln}\left[{\partial }_{\beta }⟨\text{ln}\kappa ⟩\right]$ vs $\text{ln}L$ together with its best fit function [see Eq. (14)].

Figure 14

Susceptibility ${\chi }_M$ at $n=9$ and lateral dimensions $L=24$ (diamond), 32 (circle), 40 (up triangle), 48 (square), and 60 (down triangle)