Quantification of thermal fluctuations in stripe domain patterns

T. N. G. Meier, M. Kronseder, M. Zimmermann, and C. H. Back

Phys. Rev. B 93, 064424 – Published 23 February 2016

Abstract

In ultrathin ferromagnetic films with perpendicular anisotropy a spin-reorientation transition from out-of-plane to in-plane orientation of the magnetization vector may occur. The competition of exchange and dipole interaction leads to the formation of stripe domain patterns in the vicinity of the spin reorientation transition. Here we investigate fluctuations of domain patterns in ultrathin epitaxial Ni/Fe films grown on Cu(001) using the technique of threshold photoemission magnetic circular dichroism in combination with photoemission electron microscopy allowing real-time observation of the domain pattern dynamics. The key finding of our experiments is that fluctuations can easily be quantified by calculating thermodynamic susceptibilities from a series of time resolved images. We analyze the strength of fluctuations with respect to temperature and externally applied out-of-plane magnetic fields.

Received 20 April 2015
Revised 26 January 2016

DOI:https://doi.org/10.1103/PhysRevB.93.064424

Physics Subject Headings (PhySH)

Bilayer films Ferromagnets Thin films

Reflection high-energy electron diffraction X-ray photoemission electron microscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

T. N. G. Meier, M. Kronseder, M. Zimmermann, and C. H. Back^*

Physics Department, Universität Regensburg, Universitätsstrasse 31, 93040 Regensburg, Germany

^*christian.back@ur.de

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Issue

Vol. 93, Iss. 6 — 1 February 2016

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Images

Figure 1
Linescan of the domain pattern near the SRT of a (6–12 ML)Ni/(0–6 ML)Fe/Cu(001) double-wedge-shaped sample. (a) The highly ordered stripe domain pattern is shown near the SRT. The black line separates the in-plane phase on the left from the (canted) out-of-plane phase on the right. A minimal domain width of $0.65 μ m$ is reached at the SRT. In (b) the domain width is plotted against the distance from the SRT $d_{SRT}$ . The black curve in (b) shows an exponential fit to the data. The MCD asymmetry (c) decreases almost linearly when approaching the SRT indicating a continuous SRT. (d) A phase diagram of the Ni/Fe/Cu(001) system, which is determined by tracing the SRT on a double-wedge-shaped (6–12 ML)Ni/(0–3 ML)Fe/Cu(001) sample.
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Figure 2
Calculation of the susceptibilities. This figure shows the geometric magnetization $m_{geo}$ and the exchange energy $E_{ex}$ versus time for an actual measurement, where the magnetic field is increased by a step of 0.25 Oe 2 min before recording the movie and is kept constant during the measurement. The calculation of the susceptibilities is reasonable only in periods where a stable equilibrium state is obtained (beyond the dash-dotted black line) and was done in this way throughout all calculations.
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Figure 3
Reduction of fluctuations during cooling at large domain width. The domain width (a) increases with decreasing temperature. On the left and right side of the figure representative images of the domain pattern at various temperatures are shown. The white area inside the images was not taken into account for the calculations due to a strong pinning site. The field of view of all images is $40 μ m$ . The exchange energy per area (b) is proportional to the total domain wall length and hence decreases during cooling. The residual field of $\approx 1 \times 10^{- 4} T$ in the PEEM chamber leads to a nonzero geometric magnetization, which is plotted in (c). (d) and (e) Sum images of the domain walls at 275 and 253K, respectively, reveal fluctuating areas. Red color means low occupancy of the corresponding pixel by domain walls and yellow color high occupancy. Fluctuating domain walls are therefore colored red. The inset in (d) shows the fluctuation of a domain wall between two distinct states. The variance of the total domain wall length (f) and the magnetic susceptibility (g) drastically reduce during cooling. The variance of the domain wall length is shown in two different normalizations. The red points correspond to the variance per area and the black diamonds to the variance per domain wall length.
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Figure 4
Cooling in the vicinity of the SRT. (a) The evolution of the domain width (red dots) as well as the variance of the total domain wall length (black squares) versus temperature. Images of the domain pattern at various temperatures are depicted in (b). During cooling the domain width increases at first accompanied by a strong reduction of the fluctuations of the domain walls. Below 230 K the domain pattern becomes stationary. This is explainable in terms of a strongly corrugated energy landscape, which is exemplary visualized in (c), where the free energy is plotted versus configuration coordinates $q$ of the system. If thermal energy is much lower than $Δ F_{1}$ the system gets trapped in the local minimum of the energy landscape and may stay there a long time. The system is nonergodic and samples only a smaller subset of configurations inside the local minimum if thermal energy is on the order of or larger than $Δ F_{2}$ .
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Figure 5
Application of out-of-plane magnetic fields to a (6–12 ML)Ni/(0–3 ML)Fe/Cu(001) sample. The measurement is carried out at room temperature at a Ni thickness of $\approx 9 ML$ and a Fe thickness of $\approx 1 ML$ . (a) Magnetization loop. Starting from zero applied field the initial curve (black dots) has been recorded. The subsequent down sweep of the magnetic field is plotted as red squares and the final sweep back to zero applied field as green triangles. The images on the left and right side of the figure show the domain pattern at certain values of the applied magnetic field. Note that the loop is shifted by $\approx 1 \times 10^{- 4} T$ indicating the presence of a residual field inside the PEEM chamber. The black line in the plots is a guide to the eye for zero field. In (b) and (c) the magnetic susceptibilities are plotted versus the applied magnetic field. In (b) the susceptibility is calculated from fluctuations in the domain pattern and in (c) from the derivative of the magnetization loop shown in (a) with respect to the magnetic field, which is approximated by the difference quotient. Both graphs are in good agreement and show that fluctuations in the domain pattern are maximal around zero field and are drastically reduced in high magnetic fields.
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