In-plane uniaxial magnetic anisotropy induced by anisotropic strain relaxation in high lattice-mismatched Dy/Sc superlattices

We report on the magnetic and structural characterization of high lattice-mismatched [Dy${}_{2nm}$/Sc${}_{t_{\text{Sc}}}$] superlattices, with variable Sc thickness $t_{\text{Sc}}$= 2–6 nm. We find that the characteristic in-plane effective hexagonal magnetic anisotropy K${}_6^{6,ef}$ reverses sign and undergoes a dramatic reduction, attaining values of $\approx$13–24 kJm${}^{-3}$, when compared to K${}_6^6=-0.76$ MJm${}^{-3}$ in bulk Dy. As a result, the basal plane magnetic anisotropy is dominated by a uniaxial magnetic anisotropy (UMA) unfound in bulk Dy, which amounts to $\approx$175–142 kJm${}^{-3}$. We attribute the large downsizing in K${}_6^{6,ef}$ to the compression epitaxial strain, which generates a competing sixfold magnetoelastic (MEL) contribution to the magnetocrystalline (strain-free) magnetic anisotropy. Our study proves that the in-plane UMA is caused by the coupling between a giant symmetry-breaking MEL constant M${}_{\gamma ,2}^2\approx 1$ GPa and a morphic orthorhombiclike strain ${\varepsilon }_{\gamma ,1}\approx {10}^{-4}$, whose origin resides on the arising of an in-plane anisotropic strain relaxation process of the pseudoepitaxial registry between the nonmagnetic bottom layers in the superstructure. This investigation shows a broader perspective on the crucial role played by epitaxial strains at engineering the magnetic anisotropy in multilayers.

Figure 1

Sketch of sample's rotation. (a) The magnetic field H is applied in the $c$ plane, so that c is the rotation axis and the Cartesian ($x$,$y$,$z$) axes are taken such as $z$ $\parallel$ c and, at the beginning of the torque experiment, $x$ $\parallel$ b and $y$ $\parallel$ a. (b) The longitudinal $M$${}_{\parallel }$ and transversal $M$${}_{\perp }$ components of the total magnetization M with respect to H in the rotation plane. Assuming the easy direction for $M$ is the a direction, this makes an angle $\phi$ with H, and M makes angles $\alpha$ and $\varphi$ (crystal angle) with H and the a direction, respectively. See text for further details.

Figure 2

(a) Overall view of the capacitive cell fabricated in machinable glass ceramic (Macor). (b) Side view of the capacitive cell, which shows the customized gap between the passive-fixed electrode and the active-cantilever one $a_0$. (c) Top view of the capacitive cell, which shows the active electrode (cantilever) covering the full area of the passive electrode. Briefly, a two-side polished (Macor${}^{\copyright }$) plate is cut into two pieces, where the smallest one, active block (B1), is used as support to clamp and hold the cantilever (sample) using a clamping beam. The largest, passive block (B2) is further thinned down, so that $a$${}_0$ can be customized, typically $a_0\sim 100$ $\mu$m in our capacitive cell. B1 and B2 are coated by a sputtered Al $\sim$200-nm-thick layer. The cantilever is placed face down and in electrical contact with B1; once clamped, the cantilever completely covers the sensing area, which guarantees a constant capacitance $\approx$2.25 pF at room temperature.

Figure 3

(a) X-ray diffraction intensity for a longitudinal scan of the scattering wave vector Q along the direction [00l]. Red arrows mark the superlattice reflections and the black arrow the Sc seed layer one. The line is a fit according to the model developed by Jehan et al. [56]. The best-fit parameters are displayed in Table 1. Inset shows a cross-sectional HAADF-STEM image of (11$\overline{2}$0)Al${}_2$O${}_3$/(110)Nb/(0001)Sc/[Dy${}_{22.2\text{Å}}$/Sc${}_{63.3\text{Å}}$]${}_{50}$ SL. (b) HRTEM image of the Dy$/$Sc multilayer structure close to the Sc seed layer. Dark and light contrast represent Dy and Sc layers, respectively.

Figure 4

For the Dy${}_{20}$/Sc${}_{60}$ SL. (a) Zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves. Inset graph is a magnification of the ZFC curve around $T$${}_C$. (b) Hysteresis loops at representative temperatures. Inset graph shows a magnetic phase diagram for the low-$H$ incoherent to high-$H$ coherent ferromagnetic structure. The critical fields $H$${}_{cr}$ are obtained from magnetization (full circles) and magnetoelastic stress (empty triangles) curves. In both cases, the applied magnetic field H is along the in-plane easy direction.

Figure 5

Longitudinal (black circles) $M$${}_{\mathrm{long}}$ and transversal (blue squares) $M$${}_{\mathrm{trans}}$ components of the total magnetization with respect to the applied magnetic field H in the rotation plane, i.e., (0001), as a function of the crystal angle $\varphi$ for the Dy${}_{20}$/Sc${}_{60}$ superlattice for ${\mu }_o$$H$ $=2$ T at (a) $T=20$ K and (b) $T=110$ K. (c) Temperature variation of $\Delta$$M$${}_{\parallel }$ and the calculated $\alpha$, which would reproduce $\Delta$$M$${}_{\parallel }$. The line is a mere eye guide. (d) Dependence of the experimentally determined $\alpha$ on the field angle $\phi$ at representative temperatures. For further details, see text.

Figure 6

For the $c$-oriented [Dy${}_{20}$/Sc${}_{60}$]${}_{50}$ SL. Experimental magnetic torque $L$${}_k$ as a function of the crystal angle $\varphi$ for an in-plane applied magnetic field ${\mu }_0$$H$ $=2$ T and at $T=20$ K (circles) and $T=110$ K (squares). The lines are a fit of $L$${}_k$ according to the following relationship: $L_k=2K_2^{ef}\text{sin}2\varphi +6K_6^{ef,6}\text{sin}6(\varphi +{\varphi }_o$), where ${\varphi }_o=5^{\circ }$ (continuous black line) and ${\varphi }_o=0$ (dotted red line). The best-fit parameters are $K$${}_2^{ef}=-113$ kJm${}^{-3}$ and $K$${}_6^{6,ef}=-8.3$ kJm${}^{-3}$ at $T=20$ K and $K$${}_2^{ef}=-42$ kJm${}^{-3}$ and $K$${}_6^{6,ef}=-3.1$ kJm${}^{-3}$ at $T=110$ K. Notice that $\varphi =0$ corresponds to the b direction in the hcp lattice. For further details, see text.

Figure 7

Data collected in the Dy${}_{20}$/Sc${}_{60}$ SL. (a) Extrapolation to $H$ = $\infty$ of the uniaxial $K$${}_2^{ef}$($T$,$H$) and hexagonal $K$${}_6^{ef,6}$($T$,$H$) field-dependent anisotropy constants at $T=40$ K (circles) and $T=110$ K (squares) following the standard procedure [90, 91]. (b) Temperature dependence of the uniaxial magnetic anisotropy constant $K_2^{ef}$. The continuous line is a fit according to the single-ion model [93], so that, $K$${}_2^{ef}$($T$)=$K$${}_2^{ef}$(0)${\widehat{I}}_{5/2}\left[{\mathcal{L}}^{-1}\left(m\right)\right]$ where $K$${}_2^{ef}$(0) = 175 kJm${}^{-3}$, ${\widehat{I}}_{5/2}\left[{\mathcal{L}}^{-1}\left(m\right)\right]$ is the reduced hyperbolic Bessel function, ${\mathcal{L}}^{-1}$ is the inverse Langevin function, and $m$=$M$($T$)/$M$(0) is the reduced magnetization (see top inset graph), where $M$(0) is the extrapolated to $T=0$ K for the $T$-dependent high-field magnetization $M$($T$). Bottom inset graph shows the ratio $K$=$K$${}_2^{ef}$/$K$${}_6^{ef,6}$, where the line is a linear fit, so that, $K\left(T\right)=K\left(0\right)-0.0022T$, where $K$(0)=13.56.

Figure 8

Dependence of the sixfold magnetoelastic anisotropy constant $K$${}_{6,\text{MEL}}^6$ = $K$${}_6^{6,ef}$$-$$K$${}_6^6$, on the experimental out-of-plane strain ${\varepsilon }_{\perp }$, where $K$${}_6^6=-0.76$ MJm${}^{-3}$ is the bulk Dy sixfold magnetocrystalline anisotropy constant [92] and $K$${}_6^{6,ef}$ is the measured one in the Dy/Sc SLs. The line is a mere eye guide. The inset graph shows a plot of ${\varepsilon }_{\perp }$ with the Sc layer thickness $t$${}_{\text{Sc}}$ and a linear fit ${\varepsilon }_{\perp }={\varepsilon }_{\perp }^o+\epsilon$$t$${}_{\text{Sc}}$, where the best-fitting parameter are ${\varepsilon }_{\perp }^o={\varepsilon }_{\perp }$($t$${}_{\text{Sc}})=0.55\pm 0.2$ and $\epsilon =0.33\pm 0.05$ nm${}^{-1}$. (b) Strain-dependent sixfold MEL constant $M$${}_{\alpha 2}^{66}$, where full squares are experimental values and the empty one corresponds to the estimated $M$${}_{\alpha 2}^{66}$(${\varepsilon }_{\perp }^o)=14$ MPa. The line is a nonlinear fit according to the equation $M$${}_{\alpha 2}^{66}=M_{\alpha 2}^{66}$(0)(1$+b{\varepsilon }_{\perp }$)${}^{-1}$, where the best-fit parameter is $b=3200$. (c) Variation of the uniaxial magnetic anisotropy constant $K$${}_2^{ef}$, with $t$${}_{\text{Sc}}$. The line is a linear fit, so that $K$${}_2^{ef}$ = $K$${}_2^{ef}$(0) + $k$${}_1$$t$${}_{\text{Sc}}$, where the best-fit parameters are $K$${}_2^{ef}$(0) = (1.29 $\pm$ 0.05) $\times$ 10${}^2$ kJm${}^{-3}$ and $k$${}_1$ = (0.075 $\pm$ 0.01) $\times$ 10${}^5$ GJm${}^{-2}$. For further details, see text.

Figure 9

Magnetoelastic (MEL) stress loops ${\sigma }_{\text{MEL}}$-$H$ for a Dy${}_{20}$/Sc${}_{60}$ SL at (a) $T=10$ K and (b) $T=110$ K. $\sigma$($b,a$) and $\sigma$($a,a$) correspond to the longitudinal and transversal ${\sigma }_{\text{MEL}}$ with respect to the applied magnetic field direction H $\parallel$ a. (c) Temperature dependence of the symmetry-breaking magnetoelastic (MEL) constant $M_{\gamma ,2}^2$. Lines are the best fit of $M_{\gamma ,2}^2$ for two different approaches: (1) dashed line, fit A, corresponds to the single-ion model [54], so that $M$${}_{\gamma ,2}^2$ = $M$${}_{\gamma ,2}^2$(0)${\widehat{I}}_{5/2}\left[{\mathcal{L}}^{-1}\left(m\right)\right]$ and (2) continuous line, fit B, corresponds to $M$${}_{\gamma ,2}^2$ = $M$${}_{\gamma ,2}^2\left(0\right)$m${}^2$, where $M$${}_{\gamma ,2}^2\left(0\right)=1.42$ GPa and ${\widehat{I}}_{5/2}\left[{\mathcal{L}}^{-1}\left(m\right)\right]$ is the reduced hyperbolic Bessel function, ${\mathcal{L}}^{-1}$ is the inverse Langevin function, and $m$ = $M$($T$)/$M$(0) is the reduced magnetization (see text for further details).

Figure 10

Schematic drawing of the epitaxial registry of (0001)Sc on (110)Nb according to (a) Kwo et al. [18] and (b) following the so-called Nishiyama-Wasserman [112] orientation.

Figure 11

HRTEM image of the Sc seed layer (top light gray color layer) on Nb (bottom dark gray color layer) at the $[$1$\overline{2}$10$]$ and $[$001$]$ zone-axis orientations, respectively. The bottom white dashed line marks the (110)Nb/(0001)Sc interface. The localization of a misfit edge dislocation ($b$=a${}_{\text{Sc}}/\sqrt{2}$) has been highlighted and a critical thickness of $h$${}_c\approx 7.2$ nm is observed.

Figure 12

Variation with temperature of the in-plane orthorhombiclike strain ${\varepsilon }_{\gamma ,1}$ for the [Dy${}_{20\text{Å}}$/Sc${}_{60\text{Å}}$]${}_{50}$ SL. The straight line corresponds to a linear fit, where the slope is $\approx$5$\times$10${}^{-7}$ K${}^{-1}$. Inset graph shows ${\varepsilon }_{\gamma ,1}$ for bulk Dy, obtained from existing linear magnetostriction data [55, 95].