Angular momentum compensation manipulation to room temperature of the ferrimagnet Ho$_{3-x}$Dy$_x$Fe$_5$O$_{12}$ detected by the Barnett effect

TL;DR

This study demonstrates that the angular momentum compensation temperature in Ho-Dy ferrimagnet can be tuned to room temperature using the Barnett effect, enabling potential applications in magnetic devices through simple mechanical rotation.

Contribution

It introduces a method to control and detect the angular momentum compensation temperature in Ho-Dy ferrimagnet using the Barnett effect, with the key finding that it can be set to room temperature.

Findings

$T_A$ increases with Dy content.

$T_A$ of Ho$_{1.5}$Dy$_{1.5}$Fe$_5$O$_{12}$ is at room temperature.

Barnett effect can be used to explore angular momentum in magnetic materials.

Abstract

We demonstrate that the angular momentum compensation temperature $T_A$, at which the net angular momentum in the sample disappears, can be controlled in Ho$_3$Fe$_5$O$_{12}$ by partially substituting Dy for Ho. The $T_A$ can be detected using the Barnett effect, by which mechanical rotation magnetizes an object due to spin-rotation coupling. We found that $T_A$ increases with the Dy content and clarified that the $T_A$ of Ho$_{1.5}$Dy$_{1.5}$Fe$_5$O$_{12}$ coincides with room temperature. The Barnett effect enables us to explore materials applicable to magnetic devices utilizing the angular momentum compensation only by rotating the powder sample at room temperature.

Equations

$$\mathcal{H}=-\mathbf{J}\cdot \mathbf{\Omega },$$

Full text

Angular momentum compensation is a key characteristic in the field of spintronics, where attention is focused on the high-speed magnetic response at the angular momentum compensation temperature.Binder et al. (2006); Stanciu et al. (2006, 2007); Kim et al. (2017) N-type ferrimagnets have a magnetic compensation temperature $T_{\rm M}$, at which magnetization disappears even in the ferrimagnetically ordered state. Furthermore, when $g$-factors of the magnetic moment belonging to different sublattices are different, the ferrimagnetic materials have an additional compensation, namely the angular momentum compensation temperature $T_{\rm A}$, at which the net angular momentum $\langle J_{\rm net}\rangle$ in the material disappears even in the magnetically ordered state. While $T_{\rm M}$ is easily determined by magnetization measurements, $T_{\rm A}$ cannot be determined by conventional magnetization measurements using a magnetic field. It was recently reported that the domain wall mobility was enhanced at $T_{\rm A}$ in GdFeCo.Kim et al. (2017) However, this method requires microfabrication of materials and only applies to metals. Nevertheless, using the Barnett effect, a phenomenon by which a rotating object is magnetized by spin-rotation coupling, the angular momentum compensation can be determined.Imai et al. (2018)

The Barnett effect was discovered in 1915.Barnett (1915, 1935) The angular momentum of electrons in matter interacts with rotational motion through spin-rotation coupling, described as

[TABLE]

where $\bm{J}$ is the angular momentum and $\bm{\Omega}$ is the angular velocity.de Oliveira and Tiomno (1962) The angular momentum is then aligned with the direction of the rotation axis, which causing magnetization. With ferrimagnets, ${\bm{J}}$ is represented as the sum of the angular momenta belonging to different sublattices $\langle J_{\rm net}\rangle$. The Barnett effect does not apply when $\langle J_{\rm net}\rangle=0$, which is the definition of the angular momentum compensation.

In this letter, we demonstrate $T_{\rm A}$ control for the insulating rare earth iron garnet Ho3-xDyxFe5O12. $T_{\rm A}$ is observed using the Barnett effect.Imai et al. (2018) To realize a high-speed magnetic device using the angular momentum compensation, $T_{\rm A}$ should coincide with the temperature at which the device is operated. We found that $T_{\rm A}$ coincides with room temperature (20${}^{\circ}\mathrm{C}$) at $x=1.5$.

Powder samples of Ho3-xDyxFe5O12 were prepared from Fe2O3(4N), Ho2O3(3N), and Dy2O3(3N) via a solid-state reaction. Fe2O3, Ho2O3, and Dy2O3 powders were mixed in a molar ratio of $5:3-x:x$ in an agate mortar. The mixed powder was pelletized and heated to $1200\ {}^{\circ}\mathrm{C}-1400\ {}^{\circ}\mathrm{C}$ in the ambient atmosphere. To characterize the samples, the DC-magnetization measurements were performed by extraction magnetometry with a commercial magnetometer (PPMS, Quantum Design).

Figure 1(a) shows the temperature dependence of the magnetization of Ho3-xDyxFe5O12. The magnetization of both the end materials, Ho3Fe5O12 (HoIG) and Dy3Fe5O12 (DyIG), vanishes at $T_{\rm M}$ = 135 and 218 K, respectively. For the substituted materials, the magnetization does not disappear completely due to the $T_{\rm M}$ distribution; thus $T_{\rm M}$ was defined as the temperature at which the magnetization shows a local minimum. Figure 1(b) shows the linear relation between $T_{\rm M}$ and the Dy concentration, and a schematic of the magnetization at sublattices. In the high-temperature region above $T_{\rm M}$, since the magnetization at the Fe3+ sublattice, $M_{\rm Fe}$, is larger than that at the $R^{3+}$ sublattice, $M_{R}$, $M_{\rm Fe}$ aligns parallel to the magnetic field. The magnitude of $M_{R}$ increases as the temperature decreases, and $M_{R}$ equals $M_{\rm Fe}$ at $T_{\rm M}$. The magnetization of these sublattices flips across $T_{\rm M}$, and then $M_{R}$ aligns parallel to the magnetic field below $T_{\rm M}$ (see Section I of Supplementary Material for more details).

We measured the Barnett effect to observe angular momentum compensation. Our apparatus for rotating samples uses an air-driven rotor system as shown in Fig. 1(c).Ono et al. (2015); Ogata et al. (2017a, b); Imai et al. (2018) The rotor system was placed in magnetic shields to exclude the geomagnetic field. The sample was rotated using compressed air with angular velocity $\Omega$ and magnetized to $M_{\Omega}$ by the rotation. We measured the stray field from $M_{\Omega}$ using a fluxgate magnetic sensor. To remove the residual background magnetic field, we measured the difference in the stray field $\Delta B=[B(+\Omega)-B(-\Omega)]/2$, where $B(+\Omega)$ and $B(-\Omega)$ represent the stray fields obtained at $+\Omega$ and $-\Omega$, respectively (Fig. 1(c)). The measurements were repeated over 20 times to remove fluctuation of the background magnetic field and increase data accuracy. $M_{\Omega}$ was calculated from $\Delta B$ using the dipole model described in Ref. Ogata et al., 2017a. For experimental details of Barnett effect measurement at room temperature and low temperatures, see Refs. Ogata et al., 2017a and Imai et al., 2018, respectively.

Figure 2(a) shows the temperature dependence of $M_{\Omega}$ of the Dy content for $x=0$ and 0.5 at a rotational frequency of $\Omega/2\pi$=1.5 kHz. The $M_{\Omega}$ of HoIG crosses zero at two temperatures, $T_{\rm M}=135$ K and $T_{\rm A}=240$ K.Imai et al. (2018) On Dy substitution with $x=0.5$, $T_{\rm M}$ and $T_{\rm A}$ increase to 150 and 260 K, respectively. This indicates that Dy substitution in HoIG can control $T_{\rm M}$ and $T_{\rm A}$.

In Ho3-xDyxFe5O12, the net angular momentum $\langle J_{\rm net}\rangle$ and magnetization $M_{\rm net}$ are described as $\langle J_{\rm net}\rangle=\langle J_{\rm Fe}\rangle-\langle J_{R}\rangle$ and $M_{\rm net}=M_{\rm Fe}-M_{R}$, where $\langle J_{\rm Fe}\rangle$ ($\langle J_{R}\rangle$) and $M_{\rm Fe}$ ($M_{R}$) represent the expected angular momentum and magnetization at ${\rm Fe}^{3+}$ ($R^{3+}$) sublattices, respectively (Fig. 2(b)). An effective $g$ factor is defined as $M_{\rm net}=g\langle J_{\rm net}\rangle$.

Above $T_{\rm A}$, since $\langle J_{\rm net}\rangle$ and $M_{\rm net}$ are coupled antiparallel due to the negative sign of the effective $g$ factor, $M_{\Omega}$ becomes negative as shown in Fig. 2(c), because Eq. (1) indicates that $+\Omega$ rotation aligns $\langle J_{\rm net}\rangle$ with the $+z$ direction. At $T_{\rm A}$, since the Barnett effect does not apply due to the net zero angular momentum, $M_{\Omega}$ becomes zero. In the range of temperatures between $T_{\rm A}$ and $T_{\rm M}$, $M_{\Omega}$ becomes positive. This indicates that $\langle J_{\rm net}\rangle$ and $M_{\rm net}$ are coupled parallel; the sign of the $g$ factor becomes positive: this is a gyromagnetic reversal state.Imai et al. (2018) At $T_{\rm M}$, though $\langle J_{\rm net}\rangle$ aligns with the rotation axis, $M_{\Omega}$ is zero due to disappearance of $M_{\rm net}$ at $T_{\rm M}$. Below $T_{\rm M}$, $M_{\Omega}$ becomes negative again due to the antiparallel coupling between $\langle J_{\rm net}\rangle$ and $M_{\rm net}$. The increase in $T_{\rm A}$ with increasing $x$ indicates that $T_{\rm A}$ coincide with room temperature (293 K) at appropriate Dy substitution.

To investigate the Ho3-xDyxFe5O12 composition with $T_{\rm A}=293$ K, the rotational frequency dependence of $M_{\Omega}$ was measured for various Ho3-xDyxFe5O12 compositions at 293 K as shown in Fig. 3(a). For all compositions, $M_{\Omega}$ is proportional to $\Omega$, indicating that the Barnett effect was correctly measured. In HoIG, $M_{\Omega}$ is negative and decreases as $\Omega$ increases because $\langle J_{\rm net}\rangle$ and $M_{\rm net}$ are coupled antiparallel above $T_{\rm A}=240$ K (see Fig.2(a)). As the Dy content $x$ increases, the slope of $M_{\Omega}$ vs. $\Omega$ gradually increases, becoming almost 0 at $x=1.5$, and becoming positive at $x\geq 2.0$. Figure 3(b) shows the initial susceptibility induced by rotation: $M_{\Omega}/2\pi\Omega$ at 293 K. $M_{\Omega}/2\pi\Omega$ increases with increasing Dy content $x$. From the linear fit to the data, $T_{\rm A}$ is equal to 293 K at $x=1.49\pm 0.31$.

Figure 3(c) shows a phase diagram of Ho3-xDyxFe5O12. Both $T_{\rm M}$ and $T_{\rm A}$ depend linearly on $x$. The black line is the linear fit to the $T_{\rm M}$ data. The red line is obtained by calculating the temperature dependence of $\langle J_{R}\rangle$ with various Dy contents (see Sections II and III of Supplementary Material). The calculations are consistent with our experimental results.

Here, we consider the Dy substitution effect on $T_{\rm M}$. In the general rare earth iron garnet ($R$IG) system, $T_{\rm M}$ predominantly depends on $M_{R}$, which contains two factors: the size of the $R^{3+}$ magnetic moment $\mu=-gJ$, and $R^{3+}$—Fe3+ interaction (see Section I of Supplementary Material). Since the Néel temperature is almost irrelevant to the rare earth $R^{3+}$, the ferrimagnetic order in $R$IG is predominantly contributed by the Fe3+—Fe3+ interaction. Thus, the temperature dependence of magnetization at the Fe3+ sublattice can be regarded as almost the same in the $R$IG series. The $R^{3+}$—Fe3+ interaction contributes to the temperature dependence of the magnetization at $R^{3+}$ sublattice. The $R^{3+}$—$R^{3+}$ interaction is much weaker than any other interactions in $R$IG and is negligible except in the very low temperature region. With HoIG and DyIG, since Ho3+ and Dy3+ have the same magnetic moment of $\mu=10\ \mu_{\rm B}$, $T_{\rm M}$ only depends on the $R^{3+}$—Fe3+ interaction. Figure 4 shows the calculated temperature dependence of sublattice magnetization (see Section II of Supplementary Material). Because of the stronger Dy3+—Fe3+ interaction compared to Ho3+—Fe3+, the magnetization at the Dy3+ sublattice is larger than that at the Ho3+ sublattice. Consequently, the $T_{\rm M}$ of DyIG is higher than that of HoIG.

In contrast to $T_{\rm M}$, $T_{\rm A}$ depends on $\langle J_{R}\rangle$, which contains two factors: the size of the $R^{3+}$ angular momentum $J_{R}$, and the $R^{3+}$—Fe3+ interaction. The calculated temperature dependence of the angular momenta at the ${\rm Ho}^{3+}$ and ${\rm Dy}^{3+}$ sublattices is shown in Fig. 4. Although the angular momentum of Dy3+ ($J_{\rm Dy}=15/2$) is smaller than that of Ho3+ ($J_{\rm Ho}=8$), $\langle J_{\rm Dy}\rangle$ is larger than $\langle J_{\rm Ho}\rangle$ because the Dy3+—Fe3+ interaction is stronger than the Ho3+—Fe3+ interaction. Therefore, the $T_{\rm A}$ of DyIG is larger than that of HoIG.

We now discuss the relative values of $T_{\rm M}$ and $T_{\rm A}$. Essentially, the difference between $T_{\rm M}$ and $T_{\rm A}$ originates in the difference in the $g$ factors of the ions belonging to different sublattices. Especially, when all the ions have the same $g$ factor, $T_{\rm A}$ merges with $T_{\rm M}$. When the difference in $g$ factors between Fe3+ and $R^{3+}$ increases, so does the difference between $T_{\rm M}$ and $T_{\rm A}$. In the Ho3-xDyxFe5O12 system, the $g$ factors of Fe3+, Ho3+, and Dy3+ are 2, 5/4, and 4/3, respectively. Since the difference in the $g$ factors in DyIG is smaller than that in HoIG, the difference between $T_{\rm M}$ and $T_{\rm A}$ is expected to be smaller in DyIG than in HoIG. However, the experimental result shows that $T_{\rm A}(x)$ increases at almost the same rate as $T_{\rm M}(x)$ against temperature as shown in Fig.3(c), This is due to competition between two effects of Dy substitution: $T_{\rm A}$ increases due to the stronger Dy3+—Fe3+ interaction compared to Ho3+—Fe3+, and $T_{\rm A}$ decreases relative to $T_{\rm M}$ due to the larger $g$ factor of Dy3+ compared to that of Ho3+.

In summary, we controlled the angular momentum compensation temperature of Ho3-xDyxFe5O12 by Dy substitution, and found that $T_{\rm A}$ coincides with room temperature at $x=1.5$ in this system using our apparatus for measuring the Barnett effect. Using the Barnett effect, $T_{\rm A}$ is easily obtained without microfabrication of the sample, regardless of the metal or insulator. The Barnett effect enables us to explore candidate materials for high-speed magnetic devices exploiting fast-magnetization reversal at angular momentum compensation.

Supplementary material

See supplementary material for details of the magnetic structure of the samples and their calculated magnetization and angular momentum.

Acknowledgements.

This work was supported by JST ERATO Grant Number JPMJER1402, JSPS Grant-in-Aid for Scientific Research on Innovative Areas Grant Number JP26103005, and JSPS KAKENHI Grant Numbers JP16H04023, JP17H02927.