Finite-size Kosterlitz-Thouless transition in 2DXY Fe/W(001) ultrathin films

Magnetic susceptibility measurements of 3–4 ML Fe/W(001) ferromagnetic films demonstrate that this is a 2DXY system in which a finite-size Kosterlitz-Thouless (KT) transition occurs. The films are grown in ultrahigh vacuum and their magnetic response is measured using the magneto-optic Kerr effect (MOKE). The analysis of many independently grown films shows that the paramagnetic tail of the susceptibility is described by $\chi \left(T\right)={\chi }_{0}exp[B/{(T/T_{\text{KT}}-1)}^a]$, where $a=0.50\pm 0.03$ and $B=3.48\pm 0.16$, in quantitative agreement with KT theory. Below the finite size L transition temperature $T_C\left(L\right)$, the behavior is complicated by dissipation (likely related to the re-emergence of fourfold anisotropy and magnetic domains). A subset of measurements with very small dissipation most closely represents the idealized system treated by theory. For these measurements, there is a temperature interval of order tens of K between the fitted Kosterlitz-Thouless transition temperature and the finite-size transition temperature, in agreement with theory. The normalized interval $T_C\left(L\right)/T_{\text{KT}}-1=0.065\pm 0.016$ yields an estimate of the finite size $L$ affecting the film of order micrometers. This gives experimental support to the idea that even a mesoscopic limitation of the vortex-antivortex gas results in a substantial finite-size effect at the KT transition. In contrast, fitting the paramagnetic tail to a power law, appropriate to a second-order critical transition, gives unphysical parameters. The effective critical exponent ${\gamma }_{\text{eff}}\approx 3.7\pm 0.7$ does not correspond to a known universality class, and the fitted transition temperature $T_{\gamma }$ is much further below the peak in the susceptibility than is physically reasonable.

Figure 1

(a) Regularly shaped magnetic susceptibility measurements fall into two categories, each occurring in roughly one third of the films. Type I signals (also shown in the inset) have a small magnitude and a very small imaginary component. Type II signals are an order of magnitude larger and have a large imaginary component. The measurements were made with an ac field amplitude of 0.56 Oe. (b) The magnetic susceptibility of a type II film is shown for selected values of the ac field amplitude. (c) The effect of the heating rate on the high temperature tail of $\chi \left(T\right)$ is assessed by how far down the peak the point of inflection, at $T_{\text{infl}}$, occurs. The findings for two different films are indicated by the circular and triangular symbols.

Figure 2

(a) A magnetic susceptibility of type I. The temperature range of the data that are fit to Eq. (6) is indicated by $T_{\text{min}}$ and $T_{\text{max}}$. The fitted curve is shown with a solid line, and the fitted temperature $T_{\text{KT}}$ by a dashed line. The peak of the susceptibility is indicated by a second dashed line, and labeled $T_C\left(L\right)$. (b) The fitted value of $B$ (left-hand scale, solid symbols) and the ${\chi }^2$ statistic of the fit (right-hand scale, open symbols) as a function of $T_{\text{min}}$. The dashed line shows the value of $T_{\text{min}}$ selected for the fit in (a). (c) As in (b), but for the selection of $T_{\text{max}}$. (d) The parameter $B$ found in least squares fits to the data in (a), as a function of the parameter $a$, is shown by the points. The solid line interpolates the points. A similar fitting procedure yields the interpolated dotted lines for type I measurements from seven additional films. The inset expands the region near $a=1/2$, and shows the fitted values of $B$, with error bars for $a=1/2$.

Figure 3

The data in Fig. 2 are fit to the power law in Eq. (7) appropriate for a second-order transition. (a) $\mathrm{Re}\chi \left(T\right)$ is fit using three independent parameters: ${\chi }_0$, $\gamma$, and the second-order transition temperature $T_{\gamma }$. The solid line illustrates the high quality fit, with a statistical ${\chi }^2$ that is essentially indistinguishable from the fit to KT theory. Dashed lines indicate $T_{\gamma }$ and the peak maximum at $T_C\left(L\right)$. (b) The same fit is shown on logarithmic scales, where the slope of the fitted line gives the exponent $\gamma =3.61\pm 0.08$.

Figure 4

(a) A magnetic susceptibility of type II. The temperature range used to fit to Eq. (6) is bounded by $T_{\text{min}}$ and $T_{\text{max}}$. The dashed lines indicate the fitted value of $T_{\text{KT}}$ and the maximum of the curve $T_{\text{peak}}$. The solid line shows the fit to the paramagnetic tail of the data. The inset compares the fit of the type II measurement in the main panel to a type I measurement from the same film, but along an easy axis at right angles to the first. The two fits have comparable ranges in temperature and the magnitude of the susceptibility. (b) The fitted values of $B$ as a function of chosen values of $a$ for the data in (a) are plotted as points with an interpolating solid line. A similar fitting procedure yields the interpolated dotted lines for type II measurements from seven additional films. The inset expands the region near $a=1/2$, and shows the fitted values of $B$, with error bars for $a=1/2$.