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by Rok Dittrich

The energy barrier between stable configurations determines the probability of a thermal switching event. In MRAM elements the energy barrier is determined by the shape of the element ( shape anisotropy ) and the induced crystalline anisotropy. This should guarantee a lifetime of a stored bit of about 10 years. We calculate energy barriers for elliptical MRAM elements using the nudged elastic band method.

The animations below show sequences of magnetization states when we take a "walk" trough the micromagnetic energy landscape following the minimum energy path from one energy minimum to the other. Along this walk we pass one ore more saddle points, so called "transition states", which determine the height of the energy barrier(s).

For parameters see the model.


Minimum energy paths at 1120 nm size

Three minimum energy paths for the thermal reversal of a 4 nm thin NiFeCo MRAM element are found between the two stable magnetization states. An opposing field is applied along the easy axis (= long axis of the cell). In path 1 and path 2 the magnetization stays in-plane crossing a single barrier. In path 3 a two step reversal mode is found passing a metastable state (vortex in the center).

The graph shows the dependence of the corresponding energy barriers on the external field strength. For an applied field in the range 0-15 Oe path 3 has the lowest barrier. From 15-80 Oe path 1 has the lowest barrier and above 80 Oe path 2 has the lowest barrier (see the graph). In a reference calculation standard micromagnetic dynamics is applied to determine the zero temperature switching field (quasistatic hysteresis calculation). The obtained value for "Hc" of 94 Oe agrees well with the switching field obtained when we extrapolate the energy barrier to zero height (see the graph).

field movie1 movie2 movie3 barriers

Size dependency


Energy barrier as a function of the easy-axis field strength for different particle size. The thickness (4 nm) and aspect ratio (1 : 2.5) was kept constant when varying the size. The size gives the length of the long axis of the element. The extrapolation to the zero barrier height gives a switching field that agrees with the value obtained in the dynamic calculation of the switching fields. At zero external field the energy barrier depends almost linearly on the square root of the area of the top surface of the element.

Inhomogeneous reversal modes in the single domain regime

field movie1 movie2 Magnetization configurations along the minimum energy paths for the thermal reversal of a 4 nm thin NiFeCo MRAM element of 62.5 nm size. An opposing field is applied along the easy axis (=long axis). For fields < 150 Oe the reversal mode is close to a homogeneous rotation of the magnetization. However for fields stronger than 150 Oe there is an abrupt change to an inhomogeneous reversal mode as shown here (500 Oe).

see explanation

Hext= 0 Oe Hext= 500 Oe

Stability of half selected cells


left) Array of MRAM cells with current lines. To switch the center cell a current is applied in the two red colored current lines. The resulting field (red arrow) is strong enough to switch the magnetization. Half selected cells feel a weaker field (blue arrows) either parallel or perpendicular to the easy axis. right) Energy barrier as a function of the applied field strength for the half selected cells with a size of 125 nm. The blue line corresponds to the field strength when a switching current is applied. Applying an easy-axis field is more critical for the stability than when applying a hard-axis field.


[1] Rok Dittrich, "Finite element computation of energy barriers in magnetic systems", Ph.D. thesis, Vienna University of Technology, 2003

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