Figure 1

Comparisons of $T_c\left(D\right)$ function between the model prediction in terms of Eq. (14) (solid lines) and the available experimental measurements for Fe/substrate thin films, where the parameters $D_0=\left(2h\right)∕2=h=0.2483\mathrm{nm}$ in terms of Eq. (9) with $d=2$, and ${\alpha }_i=1.612$ according to Eq. (13) with $J_s\approx J_i$, $J_{\mathrm{sub}}\approx 0$ (Ref. 40) and ${\alpha }_s=1.612$ in terms of Eq. (11). Other parameters used for calculations are listed in Table . The symbols ◆ (Ref. 2), ▷ (Ref. 3), ∎ (Ref. 4), ▵ (Ref. 5), ● (Ref. 6), and ∎ (Ref. 7) denote the experimental results of $\mathrm{Fe}∕\mathrm{Si}\mathrm{O}$, $\mathrm{Fe}∕\mathrm{Ag}\left(001\right)$, $\mathrm{Fe}∕\mathrm{Au}\left(100\right)$, $\mathrm{Fe}∕\mathrm{Pd}\left(100\right)$, $\mathrm{Fe}∕\mathrm{Ag}\left(111\right)$, and $\mathrm{Fe}∕\mathrm{Ag}\left(100\right)$ epitaxial films, respectively. Inset: the dashed, short dotted, dotted, and solid lines, respectively, denote the predictions of Eqs. (1b, 3, 4, 14). The parameters employed in the calculations are listed as: $\xi =0.447\mathrm{nm}$ in Eq. (1b) (Ref. 36), the original parameters in Eq. (3) have been cited from the original Ref. 39 with $q_i=0.8756$, 0.9376, 0.9688, $z_{\mathrm{ib}}=1∕3$, $1∕2$, $2∕3$ with $z_i=4$, 6, 8, and $z_b=12$ for $i=1$, 2, 3, respectively, which is the same for Figs. 2, 3, 4, 5, 6, 7, and $c=1∕2$ and $S_b=111.52$ $J\bullet g\text{−}{\text{atom}}^{-1}\bullet {\mathrm{K}}^{-1}$ in Eq. (4) (Ref. 38).Reuse & Permissions

Figure 2

Comparisons of $T_c\left(D\right)$ function between the model prediction in terms of Eqs. (14) (solid lines) and (15) (dash dotted lines) and the available experimental evidences for Co, ${\mathrm{Co}}_1{\mathrm{Ni}}_1$, ${\mathrm{Co}}_1{\mathrm{Ni}}_3$, and ${\mathrm{Co}}_1{\mathrm{Ni}}_9$/substrate epitaxial thin films, where the parameters in Eq. (14) $D_0=\left(2h\right)∕2=h=0.2497$, 0.2495, 0.2493, and $0.2493\mathrm{nm}$ in terms of Eq. (9) with $d=2$, and ${\alpha }_i=1.734$, 1.773, 1.792, and 1.803 according to Eq. (13) with $J_s\approx J_i$, $J_{\mathrm{sub}}\approx 0$ and ${\alpha }_s=1.734$, 1.773, 1.792, and 1.803 in terms of Eq. (11), respectively. Other parameters used for calculations are listed in Table . The parameters used in Eq. (15) are $D^{\prime }=0.195$, 0.150, $0.129\mathrm{nm}$, ${\xi }_0=0.488$, 0.570, $0.518\mathrm{nm}$, and $\lambda =1.66$, 1.49, 1.39 for ${\mathrm{Co}}_1{\mathrm{Ni}}_1$, ${\mathrm{Co}}_1{\mathrm{Ni}}_3$, and ${\mathrm{Co}}_1{\mathrm{Ni}}_9$, respectively (Ref. 10). The symbols 엯 (Ref. 8), ∗ (Ref. 9, 10), ∎ (Ref. 10), + (Ref. 11) denote the experimental evidences of $\mathrm{Co}∕\mathrm{Cu}\left(100\right)$, $\mathrm{Co}∕\mathrm{Cu}\left(111\right)$, $\mathrm{Co}∕\mathrm{Cu}\left(001\right)$ epitaxial films; ◆ (Ref. 10), ▴ (Ref. 10), and ★ (Ref. 10) denote ${\mathrm{Co}}_1{\mathrm{Ni}}_1∕\mathrm{Cu}\left(100\right)$, ${\mathrm{Co}}_1{\mathrm{Ni}}_3∕\mathrm{Cu}\left(100\right)$, ${\mathrm{Co}}_1{\mathrm{Ni}}_9∕\mathrm{Cu}\left(100\right)$ epitaxial thin films, respectively. Inset: the dashed, the short dotted, the dotted, the solid, and the dash-dotted lines denote Eqs. (1a, 1b, 3, 4, 14, 15), respectively. The parameters utilized in the calculations for Co are listed as: $\xi =0.396\mathrm{nm}$ in Eq. (1b) (Ref. 36), $c=1∕2$ and $S_b=117.62$ $J\bullet g\text{−}{\text{atom}}^{-1}\bullet {\mathrm{K}}^{-1}$ in Eq. (4) (Ref. 38), and $D^{\prime }=0.180\mathrm{nm}$, ${\xi }_0=0.324\mathrm{nm}$, and $\lambda =1.02$ in Eq. (15), respectively.Reuse & Permissions

Figure 3

Comparisons of $T_c\left(D\right)$ function between the model predictions in terms of Eq. (14) and available experimental evidences for Ni nanoparticles $(d=0)$, nanorods $(d=1)$, and thin films $(d=2)$, respectively, where the parameters $D_0=1.4952$, 0.9968, $0.2492\mathrm{nm}$ in term of Eq. (9) with $d=0$, 1, 2, and ${\alpha }_s=1.811$ in light of Eq. (11) for nanoparticles and nanorods, and ${\alpha }_i={\alpha }_s=1.811$ in terms of Eq. (13) with $J_s\approx J_i$ and $J_{\mathrm{sub}}\approx 0$ for epitaxial films on inert substrates (Ref. 40). The prediction of Eq. (2) is shown as the short dashed with $\Delta L=0.8084\mathrm{nm}$ in terms of Eq. (17). Other parameters used for calculations are listed in Table . The symbols ▾,(Ref. 10), ▴ (Ref. 10), ● (Ref. 11), ★ (Ref. 12), and ◇ (Ref. 13) denote the experimental evidences of $\mathrm{Ni}∕\mathrm{Cu}\left(111\right)$, $\mathrm{Ni}∕\mathrm{Cu}\left(100\right)$, $\mathrm{Ni}∕\mathrm{Cu}\left(001\right)$, $\mathrm{Ni}∕\mathrm{W}\left(110\right)$ epitaxial films; ◆ (Ref. 18), ∗ (Ref. 19) denote Ni nanorods; and 엯 (Ref. 19) ∎ (Ref. 20), ▿ (Ref. 39) denote Ni nanoparticles. Inset: the dashed, the short dotted, the dotted, and the solid lines denote Eqs. (1), (3), (4), and (14), respectively. The parameters utilized in the calculations are listed as: $\xi =0.846\mathrm{nm}$ and $\lambda =1$ in Eq. (1) (Ref. 36), $c=1∕2$, and $S_b=116.22$ $J\bullet g\text{−}{\text{atom}}^{-1}\bullet {\mathrm{K}}^{-1}$ in Eq. (4) (Ref. 38), respectively.Reuse & Permissions

Figure 4

Comparisons of $T_c\left(D\right)$ function between the model prediction in terms of Eq. (14) (solid lines) and available experimental results for Gd low-dimensional crystals: nanoparticles and epitaxial thin films, respectively, where the parameters $D_0=2.145$, $0.3575\mathrm{nm}$ according to Eq. (9) with $d=0$, 2, and ${\alpha }_s=1.508$ in terms of Eq. (11), ${\alpha }_i={\alpha }_s=1.508$ in terms of Eq. (13) with $J_s\approx J_i$ and $J_{\mathrm{sub}}\approx 0$ for epitaxial films on inert substrates (Ref. 40). The short dashed line denotes Eq. (2) with $\Delta L=0.7264\mathrm{nm}$ in terms of Eq. (17). Other parameters used to calculations are listed in Table . The symbols ∎ (Ref. 14), ◆ (Ref. 14), ◇ (Ref. 15), ▾ (Ref. 16) ▴ (Ref. 16), ★ (Ref. 17) denote the experimental measurements of $\mathrm{Gd}∕\mathrm{W}\left(110\right)$, $\mathrm{Gd}∕\mathrm{W}$, $\mathrm{Nb}∕\mathrm{Gd}$, and $\mathrm{Gd}∕\mathrm{Y}\left(0001\right)$ epitaxial films;   (Ref. 21) denotes Gd nanoparticles. Inset: the dashed, short dotted, dotted, and solid lines denote Eqs. (1), (3), (4), and (14) for Gd epitaxial films, respectively. The parameters used in the calculations are listed as: $\xi =2.485\mathrm{nm}$ and $\lambda =1.6$ in Eq. (1)(Ref. 36), $c=1∕2$ and $S_b=101.55$ $J\bullet g\text{−}{\text{atom}}^{-1}\bullet {\mathrm{K}}^{-1}$ in Eq. (4) (Ref. 38), respectively.Reuse & Permissions

Figure 5

Comparisons of $T_c\left(D\right)$ function between the model predictions in terms of Eq. (14) and the available experimental evidences for $\mathrm{Mn}{\mathrm{Fe}}_2{\mathrm{O}}_4$ and ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles, where the parameters $D_0=1.338$, $1.332\mathrm{nm}$ in terms of Eq. (9) with $d=0$, and ${\alpha }_s=1.8458$, 1.8458 in terms of Eq. (11). The dotted, dashed, short dashed, and solid lines denote model predictions of Eqs. (2, 3, 4, 14), respectively. The parameters utilized in the calculations: $\Delta L=0.7544$, $0.751\mathrm{nm}$ in Eq. (2), which is determined by Eq. (17), $c=1$ and $S_b=13R$, $13R$ $J\bullet g\text{−}{\text{atom}}^{-1}\bullet {\mathrm{K}}^{-1}$ (Ref. 38) in Eq. (4) for $\mathrm{Mn}{\mathrm{Fe}}_2{\mathrm{O}}_4$ and ${\mathrm{Fe}}_3{\mathrm{O}}_4$, respectively. Other parameters are listed in Table . The symbols ● (Ref. 22) and ∎ (Ref. 39) denote the experimental results of $\mathrm{Mn}{\mathrm{Fe}}_2{\mathrm{O}}_4$ and ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles, respectively.Reuse & Permissions

Figure 6

Comparisons of $T_{\mathrm{N}}\left(D\right)$ between the model predictions in terms of Eq. (14) and the available experimental evidences for CuO nanoparticles [엯 (Ref. 30), ◀ (Ref. 31), ★ (Ref. 32), and ▶ (Ref. 33)] and nanorods [● (Ref. 33)], and Ho [∎ and ◻ (Ref. 23] and NiO thin films [◆, ◇ (Ref. 28) and ▿(Ref. 29)], respectively. In light of the properties of magnetic exchange interaction of AFM, the nearest spacing of the parallel spin-spin coupling of AFM $h=2a$ with $a$ being the lattice parameter since the lattice of AFM can be considered to consist of two sublattice with opposite spin direction (Ref. 48). The parameters $D_0=2.738$, $4.107\mathrm{nm}$ for CuO nanorods and nanoparticles in terms of Eq. (9) with $d=0$, 1, and $D_0=\left(2h\right)∕2=h=0.7154$, $0.8420\mathrm{nm}$ for Ho and NiO thin films with $d=2$, respectively, and ${\alpha }_s=1.563$ for CuO in terms of Eq. (11) and ${\alpha }_i={\alpha }_s=1.561$, 1.583 for Ho and NiO thin films in terms of Eq. (13) with $J_s\approx J_i$ and $J_{\mathrm{sub}}\approx 0$ for epitaxial films on inert substrates. The dashed, short dashed, short dotted, and dotted lines, denote the predictions of Eqs. (1)–(4), respectively. Calculation parameters used are listed in Table . In addition, the parameters $\xi =3.13$, $1.4\mathrm{nm}$ and $\lambda =1.58$, 2.8 in Eq. (1a) for Ho, and NiO, $\Delta L=1.541$ in Eq. (2) for CuO, $c=1∕2$, $S_b=6.511$, $13R$, and $13R$ $J\bullet g\text{−}{\text{atom}}^{-1}\bullet {\mathrm{K}}^{-1}$ in Eq. (4) for Ho, NiO, and CuO (Ref. 38), respectively.Reuse & Permissions

Figure 7

Comparisons of $T_{\mathrm{N}}\left(D\right)$ between the model predictions in terms of Eq. (14) (solid line) and available experimental evidences for CoO thin films epitaxially grown on $\mathrm{Si}{\mathrm{O}}_2$ substrates [● and 엯 (Ref. 26 ∎ and ★ Ref. 27)], where the parameters $D_0=\left(2h\right)∕2=h=0.8520\mathrm{nm}$ in terms of Eq. (9) with $d=2$, and ${\alpha }_i=1.544$ according to Eq. (13) with $J_s\approx J_i$, $J_{\mathrm{sub}}\approx 0$ and ${\alpha }_s=1.544$ in terms of Eq. (11). Calculation parameters used are listed in Table . The short dashed, short dotted, and dotted lines denote the predictions according to Eqs. (1), (3), and (4), respectively, in which the parameters $\xi =2.1\mathrm{nm}$ and $\lambda =1.55$ (Ref. 26), $c=1∕2$ and $S_b=13R$ (Ref. 38). The dashed and dashed dotted lines, respectively, denote the model predictions in terms of Eq. (14) for CoO supported by ${\mathrm{Fe}}_3{\mathrm{O}}_4$ and NiO substrates with the parameter ${\alpha }_i=0.4139$ and 0.5544 in terms of Eq. (13) with ${\alpha }_s{J}_s∕(J_s+J_{\mathrm{sub}})\approx {\alpha }_s{T}_c\left(\infty \right)∕[T_c\left(\infty \right)+T_{c,\mathrm{sub}}\left(\infty \right)]$, which is achieved based on the mean-field approximation, $J_s\propto T_c\left(\infty \right)$ and $J_{\mathrm{sub}}\propto T_{c,\mathrm{sub}}\left(\infty \right)$ or $J_s\propto T_{\mathrm{N}}\left(\infty \right)$ and $J_{\mathrm{sub}}\propto T_{\mathrm{N},\mathrm{sub}}\left(\infty \right)$ (Ref. 45). The symbols ▵ (Ref. 28) and ◇ (Ref. 34) denote the experimental results of CoO thin films supported by ${\mathrm{Fe}}_3{\mathrm{O}}_4$ and NiO substrates, respectively.Reuse & Permissions