Two-dimensional $\mathit{XY}$-like amorphous Co${}_{68}$Fe${}_{24}$Zr${}_8$/Al${}_{70}$Zr${}_{30}$ multilayers

We present an experimental realization of a magnetic two-dimensional $\mathit{XY}$ system using amorphous materials. The classification of the dimensionality is based on the critical behavior of amorphous ${\mathrm{Co}}_{68}{\mathrm{Fe}}_{24}{\mathrm{Zr}}_8(d)/{\mathrm{Al}}_{70}{\mathrm{O}}_{30}\mathrm{\hspace{0.5em}}(20\AA )$ multilayers, where $d=11-16\AA$. Analysis of the remanent magnetization, the magnetic isotherms, the initial susceptibility, and the magnetic correlation length shows that the magnetic phase transition can be described by the 2D $\mathit{XY}$ model. The samples are not paramagnetic above the critical temperature but are characterized by local magnetic order manifested in the field and temperature dependence of the magnetization. Furthermore, an average spin-spin interaction length of 8.1 Å was estimated using the thickness dependence of the Curie temperature.

Figure 1

Measured x-ray reflectivity (black) together with simulations (red). Labels refer to the nominal CoFeZr thicknesses. The curves are offset for clarity and the vertical dashed line serves as a guide to the eye.

Figure 2

(a) The coercive field of the 12 Å sample versus temperature. (b) Magnetization of the 12 Å sample versus temperature. The solid line (red) corresponds to a fit to Eq. (1). The insets show hysteresis curves, where the applied field is $-6.5⩽{\mu }_0\mathrm{H}⩽6.5$ mT, at selected temperatures.

Figure 3

Magnetization versus reduced temperature plotted on a logarithmic scale. $\beta$ was determined by a linear fit (solid line) of the data in the interval ${10}^{-3}⩽-t⩽{10}^{-1}$, indicated by the dashed line. The data sets are offset for clarity.

Figure 4

Magnetization versus field at $T_{\mathrm{C}}$ plotted on a logarithmic scale. $\delta$ was determined by a linear fit (solid line) to the data. The data sets are offset for clarity.

Figure 5

Magnetization versus field at the temperature defined as $T^{*}$, where $\delta =15$, plotted on a logarithmic scale. The data sets are offset for clarity.

Figure 6

The real part of the ac susceptibility, from left to right the 12 Å, 13 Å, and 14 Å sample. The data is normalized to the maximum value of the susceptibility for each sample.

Figure 7

Linear fit to the initial susceptibility of the 13 Å sample to Eq. (6) (red dashed line) and Eq. (3) (black solid line); see text for details. The data points included in the fit are indicated by the dashed vertical lines.

Figure 8

$T_{\mathrm{C}}$ (open squares) and $T^{*}$ (closed triangles) versus the thickness of the magnetic layer. The right axis shows the relative Curie temperature and the solid line is a fit of Eq. (9) to $T_{\mathrm{C}}$.

Figure 9

The hysteresis loop at 114K of the 12 Å sample. The intersection of two linear fits to the magnetization at low and high applied field defines the cross-over field.

Figure 10

The top half of hysteresis loops of the 14 Å sample at four different temperatures. The solid lines (red) correspond to fits to Eq. (12).

Figure 11

Comparison between the inverse cross-over field and the magnetic moment from Langevin fits of the 14 Å sample. The COF data is rescaled with respect to the Langevin data at 237 K.

Figure 12

The correlation length as function of $(T/T_{\mathrm{KT}}-1){}^{-1/2}$ on a semi-logarithmic scale. The constant $b$ was determined by linear fits (black lines) to Eq. (11).