Analytical expression for critical frequency of microwave assisted magnetization switching

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<jats:title>Abstract</jats:title> <jats:p>The microwave-assisted switching (MAS) of magnetization in a perpendicularly magnetized circular disk is studied based on the macrospin model in a rotating frame. The analytical expression for the critical frequency of MAS is derived by analyzing the presence of a quasiperiodic mode. The critical frequency is expressed as a function of the radio frequency (rf) field <jats:italic>H</jats:italic> <jats:sub>rf</jats:sub> and the effective anisotropy field <jats:inline-formula> <jats:tex-math><?CDATA $H_{\text{k}}^{\text{eff}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="BN150061if001.gif" xlink:type="simple" /> </jats:inline-formula>. For a small rf field such that <jats:inline-formula> <jats:tex-math><?CDATA $H_{\text{rf}} \ll H_{\text{k}}^{\text{eff}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="BN150061if002.gif" xlink:type="simple" /> </jats:inline-formula>, the critical frequency is approximately equal to <jats:inline-formula> <jats:tex-math><?CDATA $(\gamma /\pi )\root 3 \of{\smash{H_{\text{k}}^{\text{eff}}H_{\text{rf}}^{2}}\mathstrut}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="BN150061if003.gif" xlink:type="simple" /> </jats:inline-formula>.</jats:p>

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