###### Abstract
force gradient \(F_{z}{{}^{\prime}}(=\partial F_{z}/\partial z=\partial^{2}(\mathbf{m}\cdot\mathbf{H})/ \partial z^{2})\) was given by the interaction between the magnetic moment of SP tip \(\mathbf{m}\) (\(=\chi\mathbf{H}\)) and the applied magnetic field \(\mathbf{H}\) in the following equation:  
\[F_{z}{{}^{\prime}}=\frac{\partial^{2}\mathbf{m}\cdot\mathbf{H}}{\partial z^{2}}=\chi \ \frac{\partial^{2}H^{2}}{\partial z^{2}}=\chi\frac{\partial^{2}}{\partial z^{2 }}\Big{(}\mathbf{H}_{0}\cos(\omega_{c}t)+\frac{\alpha}{2}\mathbf{H}_{0}\cos((\omega_ {c}-\omega_{m})t)+\frac{\alpha}{2}\mathbf{H}_{0}\cos((\omega_{c}+\omega_{m})t) \Big{)}^{2} \tag{1}\]  
By the inner products of magnetic field terms, low frequency terms of magnetic force derivative \(F_{z}{{}^{\prime}}(\omega_{m}t)\) (\(=\alpha\chi\frac{\partial^{2}H_{0}^{2}}{\partial z^{2}}\cos(\omega_{m}t)\)) and \(F_{z}{{}^{\prime}}(2\omega_{m}t)\) (\(=\frac{\alpha^{2}}{4}\chi\frac{\partial^{2}H_{0}^{2}}{\partial z^{2}}\cos( 2\omega_{m}t)\)) were generated. The value of \(F_{z}{{}^{\prime}}(\omega_{m}t)\) is larger than that of \(F_{z}{{}^{\prime}}(2\omega_{m}t)\) because of \(\alpha\leq 1\), so only the first harmonic term has been selected in this study. In the framework of A-MFM, we selected the low modulation frequency \(f_{m}(=\omega_{m}/2\pi)\) of 89 Hz to cause the frequency modulation of cantilever oscillation:  
\[z(t)\ \bigg{(}=A\sin\bigg{(}\omega_{0}t+\frac{\Delta\omega}{\omega_{m}}\cos( \omega_{m}t)\bigg{)}\approx A\sin(\omega_{0}t)+A\frac{\Delta\omega}{2\omega_{m }}\cos((\omega_{0}+\omega_{m})t)+A\frac{\Delta\omega}{2\omega_{m}}\cos((\omega _{0}-\omega_{m})t)\bigg{)} \tag{2}\]  
in the cantilever motion equation of \(m\frac{d^{2}z(t)}{dt^{2}}+m\gamma\frac{dz(t)}{dt}+(k_{0}+\Delta k\cos(\omega_ {m}t))z(t)=F_{0}\cos(\omega_{0}t)\). Here \(f_{0}(=\omega_{0}/2\pi)\) is the oscillation frequency near the resonant frequency \(f_{r}(=\omega_{r}/2\pi)\) of the cantilever. So, the \(\Delta k\cos(\omega_{m}t)\) can be \(F_{z}\) (\(\omega_{m}t\)) component of magnetic force gradient and thus, \(\Delta\omega=\frac{\omega_{m}}{m\gamma\omega_{0}}\Delta k=\frac{\omega_{m}}{m \gamma\omega_{0}}\alpha\chi\frac{\partial^{2}H_{0}^{2}}{\partial z^{2}}\) are derived. Then the magnetic energy imaging for high-frequency magnetic field near the sample surface becomes possible because the oscillation amplitude \(\mathbf{A}\) is constant in frequency modulation. The difference between the present method and previously reported HF-MFM, where soft magnetic tips were used, is that in the previous works lock-in detected the cantilever oscillation at modulation angular frequency \(\omega_{m}\) comparable with the mechanically oscillated frequency \(\omega_{0}\approx\omega_{r}\)[26, 30, 31]. In this case, it is not easy to reduce tip-sample distance during topographic and magnetic scan because the oscillation amplitude changes due to the beat vibration between \(\omega_{m}\) and \(\omega_{0}\). In addition, the SP tip can detect the wider range of magnetic field \(\mathbf{H}\) than soft magnetic tips because the magnetization of SP tip reversibly changes between low and high \(\mathbf{H}\) without hysteresis and does not saturate in large \(\mathbf{H}\). The experiment is a step towards the use of A-MFM in HF application, which is also important in microwave applications such as MAMR, in magnetic resonance [25] and spintronics [29, 32] phenomena.