Probing the Frequency Dispersion of Magnetic Permeability of a Sample during Dynamic Interaction with a Magnetized Probe

Abstract

The forces acting on a magnetic particle during nonrelativistic motion parallel to the surface of a homogeneous medium with a frequency dispersion of the magnetic permeability are considered. General expressions are obtained for the normal (attractive) and lateral (stopping) forces acting on a small dipole particle and an extended probe. It is shown that for an arbitrary orientation of the vector of the dipole magnetic moment of the particle, along with the force of attraction and the drag force, there also appears a velocity-dependent lateral force perpendicular to the velocity vector. The possibility of using these results to study the frequency-dependent magnetic permeability of nanostructured materials and films in the dynamic scanning mode of magnetic force microscopy (MCM) with magnetic probes is discussed. Numerical estimates are given for the magnitude of the expected forces, friction coefficients, and changes in the quality factor of the MFM oscillators in the case of frequency dispersion of the magnetic permeability of the relaxation type.

Appendices

APPENDIX

Appendix A

In accordance with the general method for solving differential equations of type (11), (12), it is necessary to find the sum of solutions of the corresponding homogeneous equations and partial solutions of inhomogeneous ones. In the domains z > 0 and z ≤ 0, the solutions of the homogeneous equation (d²/dz² – k²)y(z) = 0 are y(z) = C₁exp(–kz) and y(z) = C₂exp(kz), with C₁ and C₂ being arbitrary constants. A particular solution of the inhomogeneous equation (d²/dz² – k²)y(z) = f(z) (where f(z) is a known function) is found by convolving Green’s function G(z, z') with f(z): y(z) = $\int_{ - \infty }^{ + \infty } G$(z, z')f(z')dz'. Green’s function G(z, z') is a solution to the equation

$$({{d}^{2}}{\text{/}}d{{z}^{2}} - {{k}^{2}})G(z,z{\kern 1pt} ') = \delta (z - z{\kern 1pt} ')$$

(A1)

in the region –∞ < z < ∞ and reads

$$G(z,z{\kern 1pt} ') = - \frac{1}{{2k}}\exp ( - k{\text{|}}z - z{\kern 1pt} '{\text{|}}).$$

(A2)

Particular solutions of equations (11), (12), obviously, are found by simply multiplying (А2) by the corresponding coefficients on the right side of these equations. Taking this into account, the solution to (11) and (12) is

$${{A}_{x}} = \left\{ \begin{gathered} {{A}_{1}}\exp ( - kz) + i\frac{{{{{(2\pi )}}^{2}}}}{k}{{k}_{y}}{{m}_{z}}\delta (\omega - {{k}_{x}}V)\exp ( - k{\text{|}}z - {{z}_{0}}{\text{|}}),\quad z > 0 \hfill \\ {{B}_{1}}\exp (kz),\quad z \leqslant 0, \hfill \\ \end{gathered} \right.$$

(A3)

$${{A}_{y}} = \left\{ \begin{gathered} {{A}_{2}}\exp ( - kz) - i\frac{{{{{(2\pi )}}^{2}}}}{k}{{k}_{x}}{{m}_{z}}\delta (\omega - {{k}_{x}}V)\exp ( - k{\text{|}}z - {{z}_{0}}{\text{|}}),\quad z > 0 \hfill \\ {{B}_{2}}\exp (kz),\quad z \leqslant 0. \hfill \\ \end{gathered} \right.$$

(A4)

In expressions (A3), (A4), for simplicity, the indices k, ω of the Fourier components A_x, A_y of the vector potential are omitted. It should be noted that Fourier components A_x, A_y of the induced vector potential in the region z > 0 in (A3) and (A4) correspond to terms proportional to A₁, A₂. Since in the case under consideration we have A_z = 0, the gauge condition divA = 0 leads to the relations ik_xA₁ + ik_yA₂ = 0 and ik_xB₁ + ik_yB₂ = 0, whence

$${{A}_{2}} = - \frac{{{{k}_{x}}}}{{{{k}_{y}}}}{{A}_{1}},\quad {{B}_{2}} = - \frac{{{{k}_{x}}}}{{{{k}_{y}}}}{{B}_{1}}.$$

(A5)

Taking into account the relation H = curlA, the conditions for the continuity of components H_x, H_y of the magnetic field strength and the induction B_z at the boundary z = 0 are reduced to the requirement of the continuity of the quantities ∂A_y/∂z, ∂A_x/∂z, and μ(∂A_y/∂x – ∂A_y/∂y). Taking into account these relations and (A3)–(A5), we obtain Eqs. (13), (14) for the Fourier-transforms of the induced vector potential. The coefficients B₁, B₂ in (A3) and (A4), respectively, are given by

$${{B}_{1}} = - {{A}_{1}} + i\frac{{{{{(2\pi )}}^{2}}}}{k}{{k}_{y}}{{m}_{z}}\exp ( - k{{z}_{0}}),$$

(A6)

$${{B}_{2}} = {{A}_{1}}\frac{{{{k}_{x}}}}{{{{k}_{y}}}} - i\frac{{{{{(2\pi )}}^{2}}}}{k}{{k}_{x}}{{m}_{z}}\exp ( - k{{z}_{0}}),$$

(A7)

but they are not required in what follows, since only vector potential in the region z > 0 is used to find force (1). When calculating the contributions to the vector potential from the components of the magnetic moment m_x, m_y of the particle, we note that the derivatives δ'(z – z'), which are present in (7), (8), can be replaced by kδ(z – z'), after which the equations for the components of the vector potential will be completely similar to equations (11), (12). As a result, for m_x ≠ 0, m_y = m_z = 0 and A_{kω, x} = 0, the corresponding Fourier-transforms of the vector potential read

$$\begin{gathered} {{A}_{{{\mathbf{k}}\omega ,y}}}(z) = {{(2\pi )}^{2}}{{m}_{x}}\left[ {\frac{{\mu - 1}}{{\mu + 1}}\exp ( - k(z + {{z}_{0}}))} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}} + \,\,\exp ( - k\left| {z - {{z}_{0}}} \right|)} \right]\delta (\omega - {{k}_{x}}V), \\ \end{gathered} $$

(A8)

$$\begin{gathered} {{A}_{{{\mathbf{k}}\omega ,z}}}(z) = i\frac{{{{{(2\pi )}}^{2}}}}{k}{{k}_{y}}{{m}_{x}}\left[ {\frac{{\mu - 1}}{{\mu + 1}}\exp ( - k(z + {{z}_{0}}))} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}} - \,\,\exp ( - k\left| {z - {{z}_{0}}} \right|)} \right]\delta (\omega - {{k}_{x}}V). \\ \end{gathered} $$

(A9)

At m_y ≠ 0, m_x = m_z = 0 and A_{kω, y} = 0, respectively,

$$\begin{gathered} {{A}_{{{\mathbf{k}}\omega ,x}}}(z) = - \,{{(2\pi )}^{2}}{{m}_{y}}\left[ {\frac{{\mu - 1}}{{\mu + 1}}\exp ( - k(z + {{z}_{0}}))} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}} + \,\,\exp ( - k\left| {z - {{z}_{0}}} \right|)} \right]\delta (\omega - {{k}_{x}}V), \\ \end{gathered} $$

(A10)

$$\begin{gathered} {{A}_{{{\mathbf{k}}\omega ,z}}}(z) = - i\frac{{{{{(2\pi )}}^{2}}}}{k}{{k}_{y}}{{m}_{x}}\left[ {\frac{{\mu - 1}}{{\mu + 1}}\exp ( - k(z + {{z}_{0}}))} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}} - \,\,\exp ( - k\left| {z - {{z}_{0}}} \right|)} \right]\delta (\omega - {{k}_{x}}V). \\ \end{gathered} $$

(A11)