Geometry and orientation of the ferromagnetic bar on the Si substrate. 

Contexts in source publication

Context 1

... the m ជ V time dependence is identical with that of h V due to the linearity of the equation, i.e., the complex amplitude of oscillatory magnetization is given by m ជ ̇ = m ជ ϫ exp ͑ i ␻ t ͒ , Re ͑ m ជ ̇ ͒ = m ជ V . The investigated ferromagnetic metallic samples were prepared by electrochemical deposition onto Si substrates, and have the shape of rectangular bars of micron and submicron thicknesses. The materials and dimensions of the exam- ined samples are listed in Table I. The fabrication process started with the sputter deposition of a Permalloy seed layer onto a Si substrates. A positive photoresist was spin-coated onto Permalloy templates. Applying UV lithography, the photoresist layer was patterned. Before the electrodeposition was carried out, the resist at the rim of the substrate was stripped. For the plating process, a pulse current deposition was used, applying forward and reverse pulses for compositions Ni 45 Fe 55 and Ni 81 Fe 19 ͑ see Table II ͒ . For Co 35 Fe 65 , a sequence of forward pulses combined with a dwell periods without current was used. For each composition, samples with a thickness of 0.4 ␮ m, 1 ␮ m, and 2.5 ␮ m were cre- ated. After the electrodeposition, the samples were pla- narized using chemical-mechanical polishing. The thin-film process ended with an ion beam etching of the seed layer. Afterwards, the wafers were diced into chips. The example of the atomic force microscopy ͑ AFM ͒ image is shown in Fig. 1 for the Permalloy sample no. 3. The AFM analysis indicated that the average surface roughness was on the order of 6 nm in all the samples. The maximum measured difference in height for many measurements was about 20 nm. This indicates fair uniformity for samples with thickness on the order of 1 ␮ m. Due to shape anisotropy, the internal oscillatory magnetic field at any location inside the sample depends on the variable magnetization distribution. Similarly, the internal static magnetic field distribution depends on the static magnetization distribution. Thus, in terms of complex amplitudes, H ជ is written as H ជ = H ជ − N J M ជ and h ជ as h ជ = h ជ − N D m ជ , where H E 0 is the applied static biasing magnetic field, N J S is the static demagnetizing tensor, h ជ E is the applied driving magnetic field, and N J D is the dynamic demagnetizing tensor. We are assuming that it is possible to express the static demagnetizing field as − N J S M ជ 0 and the dynamic demagnetizing field as − N J D m ជ in terms of these demagnetizing 14 tensors. Such an approach follows Kittel, except that the components of the demagnetizing tensor of the rectangular bars are averaged values 20 ͑ i.e., the internal field inside the bar is nonuniform ͒ , and N J S is not identical to N J D due to the skin effect in conducting metallic ferromagnets. The typical skin depth in high conductivity ferromagnets at microwave frequencies is on the order of 0.1 ␮ m, and the dynamic demagnetizing field quickly decays and can be considered neg- ligible far from the surface ͑ a few skin depths ͒ . The interaction between the high-frequency magnetic field and the ferromagnetic metallic material thus occurs mainly close to the surface. In the case of in-plane H ជ E 0 the internal biasing field H ជ 0 increases close to the surface 20 as well as the magnitude of h ជ . The ferromagnetic bars therefore have a higher dynamic demagnetizing field induced by the higher driving field in that surface layer, and a smaller average static demagnetizing field in the interaction region compared to non- conductive samples with the same dimensions. The orientation of the ferromagnetic bars with respect to the biasing magnetic field in all experiments is shown in Fig. 2. The z -axis of each bar was along the signal line of the coplanar waveguide in the experimental setup described later. That way, the biasing static magnetic field coincided with the direction of propagation of the microwave, and a microwave magnetic field was generated in xy -plane due to the high-frequency current. For the microwave frequencies ͑ 0–40 GHz ͒ used here, the dimensions of the samples and the width of the signal line are substantially smaller than the electromagnetic wavelength in air or in the coplanar waveguide. One can therefore neglect the nonuniform distribution of high-frequency current in the signal line under the sample as well as the corresponding nonuniform distribution of the high-frequency magnetic field interacting with the sample. In the coordinate system depicted in Fig. 2, the tensors and vectors from Eq. ͑ 4 ͒ are written as N X 0 0 N 11 0 0 N J S = 0 N Y 0 , N J D = 0 N 22 0 , 0 0 N 0 0 ...

Context 2

... the m ជ V time dependence is identical with that of h V due to the linearity of the equation, i.e., the complex amplitude of oscillatory magnetization is given by m ជ ̇ = m ជ ϫ exp ͑ i ␻ t ͒ , Re ͑ m ជ ̇ ͒ = m ជ V . The investigated ferromagnetic metallic samples were prepared by electrochemical deposition onto Si substrates, and have the shape of rectangular bars of micron and submicron thicknesses. The materials and dimensions of the exam- ined samples are listed in Table I. The fabrication process started with the sputter deposition of a Permalloy seed layer onto a Si substrates. A positive photoresist was spin-coated onto Permalloy templates. Applying UV lithography, the photoresist layer was patterned. Before the electrodeposition was carried out, the resist at the rim of the substrate was stripped. For the plating process, a pulse current deposition was used, applying forward and reverse pulses for compositions Ni 45 Fe 55 and Ni 81 Fe 19 ͑ see Table II ͒ . For Co 35 Fe 65 , a sequence of forward pulses combined with a dwell periods without current was used. For each composition, samples with a thickness of 0.4 ␮ m, 1 ␮ m, and 2.5 ␮ m were cre- ated. After the electrodeposition, the samples were pla- narized using chemical-mechanical polishing. The thin-film process ended with an ion beam etching of the seed layer. Afterwards, the wafers were diced into chips. The example of the atomic force microscopy ͑ AFM ͒ image is shown in Fig. 1 for the Permalloy sample no. 3. The AFM analysis indicated that the average surface roughness was on the order of 6 nm in all the samples. The maximum measured difference in height for many measurements was about 20 nm. This indicates fair uniformity for samples with thickness on the order of 1 ␮ m. Due to shape anisotropy, the internal oscillatory magnetic field at any location inside the sample depends on the variable magnetization distribution. Similarly, the internal static magnetic field distribution depends on the static magnetization distribution. Thus, in terms of complex amplitudes, H ជ is written as H ជ = H ជ − N J M ជ and h ជ as h ជ = h ជ − N D m ជ , where H E 0 is the applied static biasing magnetic field, N J S is the static demagnetizing tensor, h ជ E is the applied driving magnetic field, and N J D is the dynamic demagnetizing tensor. We are assuming that it is possible to express the static demagnetizing field as − N J S M ជ 0 and the dynamic demagnetizing field as − N J D m ជ in terms of these demagnetizing 14 tensors. Such an approach follows Kittel, except that the components of the demagnetizing tensor of the rectangular bars are averaged values 20 ͑ i.e., the internal field inside the bar is nonuniform ͒ , and N J S is not identical to N J D due to the skin effect in conducting metallic ferromagnets. The typical skin depth in high conductivity ferromagnets at microwave frequencies is on the order of 0.1 ␮ m, and the dynamic demagnetizing field quickly decays and can be considered neg- ligible far from the surface ͑ a few skin depths ͒ . The interaction between the high-frequency magnetic field and the ferromagnetic metallic material thus occurs mainly close to the surface. In the case of in-plane H ជ E 0 the internal biasing field H ជ 0 increases close to the surface 20 as well as the magnitude of h ជ . The ferromagnetic bars therefore have a higher dynamic demagnetizing field induced by the higher driving field in that surface layer, and a smaller average static demagnetizing field in the interaction region compared to non- conductive samples with the same dimensions. The orientation of the ferromagnetic bars with respect to the biasing magnetic field in all experiments is shown in Fig. 2. The z -axis of each bar was along the signal line of the coplanar waveguide in the experimental setup described later. That way, the biasing static magnetic field coincided with the direction of propagation of the microwave, and a microwave magnetic field was generated in xy -plane due to the high-frequency current. For the microwave frequencies ͑ 0–40 GHz ͒ used here, the dimensions of the samples and the width of the signal line are substantially smaller than the electromagnetic wavelength in air or in the coplanar waveguide. One can therefore neglect the nonuniform distribution of high-frequency current in the signal line under the sample as well as the corresponding nonuniform distribution of the high-frequency magnetic field interacting with the sample. In the coordinate system depicted in Fig. 2, the tensors and vectors from Eq. ͑ 4 ͒ are written as N X 0 0 N 11 0 0 N J S = 0 N Y 0 , N J D = 0 N 22 0 , 0 0 N 0 0 ...