6. Negative imaginary parts of permeability (μ″ <sup>&</sup>lt; 0)

As stated before, high-frequency permeability of Fe-based conducting nanostructured materials can be tailored by shapes, phase transitions, coating and size distributions. Most importantly, all imaginary parts of permeability are positive (μ″ <sup>&</sup>gt; 0), which are accepted as an unalterable natural principle. This is correct when there are no other excitations except AC magnetic field based on the fact that positive <sup>μ</sup>″ manifests energy dissipation of magnetization precession. What happens to the imaginary parts of permeability if there is another excitation, which can compensate the energy loss during the precession? Furthermore, is it possible to tailor the high-frequency permeability by electric current? If possible, it will be possible for us to design intelligent electromagnetic devices. In this section, we propose for the first time an approach to accomplish such a goal via spin transfer torque (STT) effect. According to STT effect, when polarized electrons flow through zone of metallic ferromagnetic material with nonuniformly oriented magnetic moments (e.g., magnetic domain walls), the spins of electrons will exert torques on them, which will change the dynamic responses of precession of magnetic moments [13]. When STT is strong enough, it can even switch the direction of magnetic moments. Here we only demonstrate an example using micromagnetic simulation; our detailed results can be found in Ref. [12]. The object of micromagnetic simulations is an isolated Fe nanowire with a length of 400 nm (set as "x" axis) and the diameter of 10 nm, as shown in Figure 16. The dynamic response of magnetization (a group of magnetic moments in a direction, also can be called "macro spin") can be simulated under two external excitations: AC magnetic field and electrically polarized current. The AC magnetic field is applied along the "z" axis, and polarized electrons are flowing along the "x" axis. The software of micromagnetic simulation is a three-dimensional object-oriented micromagnetic framework (OOMMF). The dynamic responses of precession are simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation as a function of time. When the STT effect is incorporated, the modified LLG equation is expressed as [13, 14]:

$$\frac{d\overrightarrow{\boldsymbol{M}}}{dt} = \chi\overrightarrow{\boldsymbol{H}}\_{\text{eff}} \times \overrightarrow{\boldsymbol{M}} + \frac{a}{\boldsymbol{M}s} \left(\overrightarrow{\boldsymbol{M}}\_{\text{s}} \times \frac{d\overrightarrow{\boldsymbol{M}}}{dt}\right) - \left(\overrightarrow{\boldsymbol{u}} \cdot \overrightarrow{\boldsymbol{\nabla}}\right)\overrightarrow{\boldsymbol{M}} + \frac{\rho}{\boldsymbol{M}s}\overrightarrow{\boldsymbol{M}} \times \left[\left(\overrightarrow{\boldsymbol{u}} \cdot \overrightarrow{\boldsymbol{\nabla}}\right)\overrightarrow{\boldsymbol{M}}\right] \tag{12}$$

where α (damping constant) is set as 0.01 for the simulations; γ is the Gilbert gyromagnetic ratio (2.21 � <sup>10</sup><sup>5</sup> mA�<sup>1</sup> <sup>S</sup>�<sup>1</sup> ); M is the magnetization; Heff is the

Figure 16. Inhomogeneous orientation of magnetic moments of an isolated nanowire (copyright, 2018, IOP).

effective magnetic field which consists of demagnetization field, exchange interaction, anisotropic field and external applied field; and finally, β (nonadiabatic spin transfer parameter) is set as 0.02 and is used to consider the impact of temperature on the dynamics of precession. The vector u is defined as

$$
\overrightarrow{\mu} = -\frac{\text{g}\mu\_B P}{2e\text{M}\_s}\overrightarrow{J}\tag{13}
$$

where g is the Landé factor, e is the electron charge, μ<sup>B</sup> is the Bohr magneton, J is the current density and P is the polarization ratio of current and set as 0.5. The simulation parameters for iron (Fe) nanowire are Ms = 17 � <sup>10</sup><sup>5</sup> A/m, magnetocrystalline anisotropy constant, K1 = 4.8 � <sup>10</sup><sup>4</sup> J/m<sup>3</sup> , exchange stiffness constant and <sup>A</sup> = 21 � <sup>10</sup>�<sup>12</sup> J/m. Also, the nanowire is discretized into many tetrahedron cells (cell size: 5 � 1.25 � 1.25 nm). The cell is smaller than the critical exchange length. To obtain high-frequency permeability spectra, these steps are followed: Firstly, the remanent magnetization state should be acquired after the external field is removed. Secondly, a pulse magnetic field is applied perpendicular to the long axis (x-axis) of the wire. The pulse magnetic field has the form h(t) = 1000 exp(�10<sup>9</sup> t) (t in second, hour in A/m). A polarized current (density J is 3.0 � <sup>10</sup><sup>12</sup> A/m<sup>2</sup> ) is flowing along x-axis. Thirdly, the dynamic responses of magnetization in time-domain are recorded under these two excitations. Using the Fast Fourier Transform (FFT) technique, the dynamic responses in frequency domain are obtained. The high-frequency permeability spectra are calculated based on the definition of permeability:

$$\mu(f) = \mathbf{1} + \frac{m(f)}{h(f)} = \mathbf{1} + \chi'(f) - i\chi''(f) \tag{14}$$

spectrum is naturally found to be different and is illustrated in Figure 17c. The previous minor resonance becomes the major resonance with negative <sup>μ</sup>″ values. The "My" component does not vanish gradually; see Figure 17d. According to the STT effect, a spin-polarized current flows through the nanowire; STTs only act on the magnetic moments at the ends of nanowire. These STTs will counteract the torques due to effective magnetic field, which will then bring the magnetizations back to their equilibrium positions. When the STT is strong enough, it can maintain the precession angle (μ″ = 0) and even enlarge the precession angle (μ″ <sup>&</sup>lt; 0) or switch the direction of magnetization. Additionally, negative <sup>μ</sup>″ values are found only for "edge mode" at lower frequency. This is because the STT effect only acts on magnetic moments with a spatially nonuniform orientation, which are only found at both ends of nanowires. The cause for minor peak becoming major peak under STT effect is due to the fact that the spin transfer torques are consistently acting on

The variation of total energy without and with STT excitation (copyright, 2018, Elsevier).

Permeability spectra and responses of My component without and with STT excitation (J 6¼ 0) (copyright,

High-Frequency Permeability of Fe-Based Nanostructured Materials

DOI: http://dx.doi.org/10.5772/intechopen.86403

Figure 17.

Figure 18.

133

2018, Elsevier).

where m(f) is magnetization under both excitations and the pulse magnetic field (h) works as the perturbing field. The following relations exist: μ<sup>0</sup> = 1 + χ<sup>0</sup> (f), <sup>μ</sup>″ <sup>=</sup> <sup>χ</sup> (f). The differences in the high-frequency permeability under different excitations have been investigated.

When single nanowire is under the excitation of only an AC magnetic field (h), two Lorentzian-type resonance peaks are found in the permeability spectrum, as shown in Figure 17a. One is located approximately at 18 GHz, and the other is located approximately at 31.5 GHz. According to our previous studies, the major resonance is called "bulk mode," which has larger magnetic loss and is manifested by the larger <sup>μ</sup>″ value, which is ascribed to the precession of perfectly aligned magnetic moments within the nanowire body excluding the misaligned magnetic moments at both ends of nanowire. The minor resonance peak located at 18 GHz is often called "edge mode," whose smaller magnitudes of real and imaginary parts are due to smaller volume fractions of magnetic moments at the ends of nanowire. The edge mode is due to the precession and resonance of misaligned magnetic moments. Physically, if external field is not applied, the magnetizations are aligned along the local effective field (Heff). Once the perturbation field (h) is applied to excite the precession, M will move away from their equilibrium positions with small angles, and M will then precess around Heff. When precession goes on, the y-component of M (My) will have nonzero values, as shown in Figure 17b. During the period of precession, the energy absorbed from the AC magnetic field will be gradually dissipated via damping mechanisms. M will restore to its equilibrium positions, and My will be zero; see Figure 17b. The energy loss is manifested by a positive imaginary part of permeability (μ″ <sup>&</sup>gt; 0). When the nanowire is excited under both the pulse magnetic field and polarized current, the high-frequency permeability

High-Frequency Permeability of Fe-Based Nanostructured Materials DOI: http://dx.doi.org/10.5772/intechopen.86403

effective magnetic field which consists of demagnetization field, exchange interaction, anisotropic field and external applied field; and finally, β (nonadiabatic spin transfer parameter) is set as 0.02 and is used to consider the impact of temperature

> !¼ � <sup>g</sup>μBP 2eMs J !

the current density and P is the polarization ratio of current and set as 0.5. The

constant and <sup>A</sup> = 21 � <sup>10</sup>�<sup>12</sup> J/m. Also, the nanowire is discretized into many tetrahedron cells (cell size: 5 � 1.25 � 1.25 nm). The cell is smaller than the critical exchange length. To obtain high-frequency permeability spectra, these steps are followed: Firstly, the remanent magnetization state should be acquired after the external field is removed. Secondly, a pulse magnetic field is applied perpendicular to the long axis (x-axis) of the wire. The pulse magnetic field has the form

netization in time-domain are recorded under these two excitations. Using the Fast Fourier Transform (FFT) technique, the dynamic responses in frequency domain are obtained. The high-frequency permeability spectra are calculated based on the

h f ð Þ <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>0</sup>

(f). The differences in the high-frequency permeability under different excitations

(h) works as the perturbing field. The following relations exist: μ<sup>0</sup> = 1 + χ<sup>0</sup>

where m(f) is magnetization under both excitations and the pulse magnetic field

When single nanowire is under the excitation of only an AC magnetic field (h), two Lorentzian-type resonance peaks are found in the permeability spectrum, as shown in Figure 17a. One is located approximately at 18 GHz, and the other is located approximately at 31.5 GHz. According to our previous studies, the major resonance is called "bulk mode," which has larger magnetic loss and is manifested by the larger <sup>μ</sup>″ value, which is ascribed to the precession of perfectly aligned magnetic moments within the nanowire body excluding the misaligned magnetic moments at both ends of nanowire. The minor resonance peak located at 18 GHz is often called "edge mode," whose smaller magnitudes of real and imaginary parts are due to smaller volume fractions of magnetic moments at the ends of nanowire. The edge mode is due to the precession and resonance of misaligned magnetic moments. Physically, if external field is not applied, the magnetizations are aligned along the local effective field (Heff). Once the perturbation field (h) is applied to excite the precession, M will move away from their equilibrium positions with small angles, and M will then precess around Heff. When precession goes on, the y-component of M (My) will have nonzero values, as shown in Figure 17b. During the period of precession, the energy absorbed from the AC magnetic field will be gradually dissipated via damping mechanisms. M will restore to its equilibrium positions, and My will be zero; see Figure 17b. The energy loss is manifested by a positive imaginary part of permeability (μ″ <sup>&</sup>gt; 0). When the nanowire is excited under both the pulse magnetic field and polarized current, the high-frequency permeability

where g is the Landé factor, e is the electron charge, μ<sup>B</sup> is the Bohr magneton, J is

t) (t in second, hour in A/m). A polarized current (density J is

) is flowing along x-axis. Thirdly, the dynamic responses of mag-

(13)

(f), <sup>μ</sup>″ <sup>=</sup> <sup>χ</sup>

, exchange stiffness

ð Þ� f iχ}ð Þf (14)

u

simulation parameters for iron (Fe) nanowire are Ms = 17 � <sup>10</sup><sup>5</sup> A/m,

magnetocrystalline anisotropy constant, K1 = 4.8 � <sup>10</sup><sup>4</sup> J/m<sup>3</sup>

<sup>μ</sup>ð Þ¼ <sup>f</sup> <sup>1</sup> <sup>þ</sup> m f ð Þ

h(t) = 1000 exp(�10<sup>9</sup>

definition of permeability:

have been investigated.

132

3.0 � <sup>10</sup><sup>12</sup> A/m<sup>2</sup>

on the dynamics of precession. The vector u is defined as

Electromagnetic Materials and Devices

Figure 17. Permeability spectra and responses of My component without and with STT excitation (J 6¼ 0) (copyright, 2018, Elsevier).

spectrum is naturally found to be different and is illustrated in Figure 17c. The previous minor resonance becomes the major resonance with negative <sup>μ</sup>″ values. The "My" component does not vanish gradually; see Figure 17d. According to the STT effect, a spin-polarized current flows through the nanowire; STTs only act on the magnetic moments at the ends of nanowire. These STTs will counteract the torques due to effective magnetic field, which will then bring the magnetizations back to their equilibrium positions. When the STT is strong enough, it can maintain the precession angle (μ″ = 0) and even enlarge the precession angle (μ″ <sup>&</sup>lt; 0) or switch the direction of magnetization. Additionally, negative <sup>μ</sup>″ values are found only for "edge mode" at lower frequency. This is because the STT effect only acts on magnetic moments with a spatially nonuniform orientation, which are only found at both ends of nanowires. The cause for minor peak becoming major peak under STT effect is due to the fact that the spin transfer torques are consistently acting on

Figure 18. The variation of total energy without and with STT excitation (copyright, 2018, Elsevier).

the magnetic moments; therefore, the precession angle is enlarging, and as a result, My component is also increased. From the perspective of total energy (Etotal) of magnetic moment ensemble, when no STT effect is involved, Etotal first increases due to the excitation of pulse magnetic field and then gradually attenuates due to energy loss via damping mechanisms. However, when STT effect exists, Etotal is not attenuated to a constant; see Figure 18.

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particles on their microwave

Finally, it should be pointed out that although others have also reported negative <sup>μ</sup>″ in other materials, there are no convincing physical mechanisms that have been provided to support their results (μ″ <sup>&</sup>lt; 0), and negative <sup>μ</sup>″ is mostly likely due to measurement errors.
