Temperature dependence of the exchange stiffness in FePd(001) thin films: Deviation from the empirical law $A\left(T\right)\propto M_S^2$ at intermediate temperatures

We study the temperature dependence of the exchange stiffness $A\left(T\right)$ of a thin FePd(001) film, which exhibits well-ordered magnetic stripe domains. We employed two different methods: (i) applying an exact micromagnetic model of the nucleation point and (ii) a detailed analysis of the temperature dependence of the saturation magnetization. The experimental data needed as input for the theoretical model are the stripe domain width and the nucleation field, and these were obtained as a function of temperature using soft x-ray magnetic scattering and superconducting quantum interference device (SQUID) magnetometry, respectively. The temperature independence of the domain width is very important from the point of view of the energetics of the FePd film, i.e., the number of magnetic domains remains constant per unit area. From this experimental finding we infer that the exchange stiffness must vary as a function of the temperature. We show that the $A\left(T\right)$ dependence obtained with the two procedures are consistent but different from the phenomenological law $A\left(T\right)\propto M_S^2\left(T\right)$, normally assumed to be valid in the temperature range examined $\left(T=50–250\text{K}\right)$, an intermediate temperature range for which there is no known expression of the magnetization as a function of the temperature. We have also investigated the $\text{Fe}3s$ core level and valence band of FePd using hard x-ray photoemission spectroscopy. The $\text{Fe}3s$ spectra exhibit negligible changes in the temperature range investigated, while small changes occurring over large energy scales are observed in the valence-band spectra. Based on the lack of consistency of the magnetometry and scattering results compared to local-moment theories, these results favor an itinerant magnetism picture for FePd.

Figure 1

Saturation magnetization as a function of the temperature measured by a SQUID magnetometry. The data have been normalized to the magnetization at the absolute zero, $M_S\left(0\right)=1013\text{emu}/{\text{cm}}^3$. In the inset the reduced magnetization is plotted versus $T^{3/2}$; a guide to the eye dashed line simulates the Bloch law for a constant exchange stiffness and the deviation from the experimental data in the intermediate temperature regime.

Figure 2

(Color online) In the top panel, four hysteresis cycles are shown, after correcting for the MgO substrate diamagnetism. Although looking quite similar, clear differences can be found in the value of the saturation magnetization at the different temperatures. In the bottom panel, a zoom of the hysteresis cycles in the region of the nucleation point is shown. The nucleation point is shown by a filled dot and it changes as a function of the temperature. An artificial offset, as indicated in the label, has been added to each curve for clarity.

Figure 3

(Color online) MFM image of the FePd thin film measured in air, the image is $8\times 4\mu {\text{m}}^2$. The yellow and brown stripes can be identified as the magnetic domains. From the size of the image it is possible to see that the domains are well ordered beside the surface roughness and the topological defects (white spots). From a line profile of this image, we were able to measure the lateral resolution to be 30 nm, roughly as wide as the cantilever tip. This does not forbid a clear determination of the domain width as the distance between two adjacent maxima or minima of the intensity.

Figure 4

(Color online) (a) Example of rocking curve measured to determine the stripe domain width at room temperature. The central specular peak, occurring when the incidence angle $\theta$ equals half of the detection angle $2\theta$, the diffuse scattering background, due to the surface imperfections, and at wider angles the magnetic scattering satellites giving the stripe domain are explained in the text. (b) The $q_z$ dependence of one of the magnetic scattering peak: the peak position is constant, while the diffuse scattering is strongly reduced at high incidence angles. The data acquisition time of the curve at $2\theta =50°$ is the same as the other ones, but multiplied later by a factor of 10 so as to compare to the other two. We want to stress on the high quality of the experimental data: Even at $\theta =25°$ incidence angle the magnetic scattering peak is clearly observed and its intensity is comparable to the specular one. The noise is also very low, despite the smallness of the signal, $\approx 0.1\text{pA}$ at the background level.

Figure 5

Temperature dependence of the domain width as measured by SXMRS. As seen, $W$ is temperature independent and the average domain width is $W=46\pm 3\text{nm}$.

Figure 6

(Color online) In the upper part we present the HXPES data taken at two different temperatures and referred to the Fermi level of a polycrystalline Au sample. As noticed FePd at room temperature has a higher DOS close to the Fermi level than at $T=50\text{K}$, while in the $d$-band complex the situation reverses. From both the curves an integral background has been subtracted and both have been divided by their respective integral from $+1$ to $-7.5\text{eV}$ for a common normalization. The blue waterfall line at the bottom shows the difference between the two curves obtained after the normalization procedure described in the text was applied. The lower part, taken from Garcia et al. (Ref. 15), shows the spin-resolved DOS for an ordered FePd bulk calculated in the local spin density approximation for the magnetization vector aligned along the (100) axis.

Figure 7

(Color online) Fits of the reduced magnetization data (black line and crosses) of Fig. 1 using different model functions and choice of ranges and parameters: (red) Bloch law at low-$T$ region only (range delimited by the red vertical bars), (blue) Bloch law for the whole data set and $M_S\left(0\right)=1$, and (green) whole data set and all the parameters left free to vary; (ochre) Kuz’min (Ref. 26) formula with $\beta =1/3$ (with $s=0.499\pm 0.005$) and (purple) $\beta =0.369$ (with $s=0.404\pm 0.004$). For the last two fits the reduced magnetization at $T=0\text{K}$ has been forced to one, the Curie temperature fixed to be $T_C=690{\text{K}}^4$, and $p=5/2$. A significant improvement of the ${\chi }^2$ of the fit can be obtained leaving the Curie temperature free to vary, but a value of $T_C=620\text{K}$ is obtained. Such a value is unphysical since clear magnetic peaks could be obtained at higher temperatures and the result of the fit with free parameters must be rejected.

Figure 8

(Color online) Comparison between the exchange stiffness temperature dependence obtained by the two methods described in the text.

Figure 9

(Color online) Zoom at the Fermi level of the HXPES data. The solid curves represent the data divided by the Fermi–Dirac function, while the dashed ones report the data before the division. Such procedure shows that the intensity up to $\approx 1.5\text{eV}$ binding energy is higher for the high-temperature curve, which is in agreement with the Stoner model. The abrupt increase at higher positive energies is artificial and should not be taken into account.

Figure 10

(Color online) HXPES spectrum of the $3s$ core levels of iron [overlapping to the $\left(\text{Pd}4s\right)$] at two different temperatures. To the experimental data an integral background has been subtracted and the binding energy has been referenced to the spectrum of a polycrystalline Au sample measured at the same temperature of the core-level spectra. The small peaks at 99 and 103 eV are shallow impurity levels.