Spin Pumping from Lu₃Fe₅O₁₂

Microwave spin pumping from ferromagnetic lutetium iron garnet (Lu₃Fe₅O₁₂) has been theoretically and experimentally investigated. The magnetization vector precession excited by a microwave magnetic field is transformed into a dc voltage due to the inverse spin Hall effect in the lutetium iron garnet/heavy metal heterostructure (Lu₃Fe₅O₁₂/Pt). In the experiments carried out, the external magnetic field has been varied from 0 to 6 kOe, thus making it possible to tune the resonant frequency in wide ranges. The experimental sensitivity of this heterostructure is 8.2 µV/W. A change in the dc voltage sign with a change in the magnetic field direction confirms the generation of spin current in the Lu₃Fe₅O₁₂/Pt heterostructure. The results obtained make a significant contribution to insight into spin pumping physics and may be useful for the development of new highly sensitive tunable spintronic devices.

The spin degree of freedom of electrons can be used to design new-generation data transmitters, receivers, and processors [1–4]. A key feature of these devices is that they will be based on the spin current, which does not induce Joule heating, rather than the charge current having such a drawback [5]. One of the most popular methods to generate a spin current in magnetic materials is spin pumping [6, 7]. Spin pumping is a process of transferring the spin from ferromagnetic materials to nonmagnetic ones via interface interactions [8, 9]. This phenomenon, caused by unique quantum-mechanical properties of electron spins, opens new possibilities for analyzing the fundamental problems of spin–charge interaction and developing innovative technologies relating to information carriers [10–12]. Experimental observation of spin pumping is generally based on achieving ferromagnetic resonance (FMR), which is accompanied by synchronized precession of spins in a ferromagnet under the effect of an external magnetic field [8, 10, 13]. This coordinated spin dynamics provides conditions for spin transfer through the ferromagnetic–nonmagnetic interface. As a result, the spin is transferred to the nonmagnetic material, and the spin current arises. Investigation of this current in ferromagnetic materials is of great scientific interest [3]. Researchers in the field of modern spintronics are focused on finding materials whose properties and characteristics are suitable for the practical applications such as microwave devices, weak-signal detectors, magnetic random access memory (MRAM) devices, and spin logic devices [2–4]. Experimental methods for studying spin pumping (including techniques for measuring the spin current and FMR spectra) give a more complete picture for understanding this phenomenon [14, 15]. The control of this process may play a decisive role in the development of next-generation electronic devices combining the classical and quantum technologies [12, 16]. One of the popular magnetic materials is yttrium iron garnet, which is widely used in both radio-engineering and optical applications due to its unique properties, such as low internal loss and high striction [17, 18]. In this context, it is necessary to investigate various structures based on ferrite garnets to improve some properties and parameters of devices under development. The purpose of this study is to analyze spin pumping from Lu₃Fe₅O₁₂. To this end, Lu₃Fe₅O₁₂ thin films were grown by chemical vapor deposition and experiments were carried out to observe FMR and record the voltage induced by the inverse spin Hall effect. To determine the parameters of the Lu₃Fe₅O₁₂/Pt heterostructure, the experimental data were approximated by theoretical relations.

A Lu₃Fe₅O₁₂ thin film was grown on a Gd₃Ga₅O₁₂ (111) single-crystal substrate by the metal-organic chemical vapor deposition (MOCVD) method [19, 20] using a system, the schematic of which is presented in Fig. 1. The Gd₃Ga₅O₁₂ (111) substrate was chosen because this material is paramagnetic at room temperature and does not contribute to measurements. A small permanent magnet vibrating in an ac magnetic field initiates the motion of an Eppendorf microtube containing a mixture of solid readily sublimating metal-organic compounds (precursors), as a result of which microportions of the latter are fed to a heated evaporator. The formed precursor vapor is transferred by a carrier gas (Ar) flow via hot transport lines into a vertical quartz reactor heated by an external furnace. Oxidative thermolysis of precursors with the formation of an oxide film occurs on the substrate located in the region of the maximum reactor temperature. Dipivaloylmethanates Lu(thd)₃ and Fe(thd)₃ (thd is the 2,2,6,6-tetramethylheptane-3,5-dionate anion) served as precursors. The temperatures of the hot lines and the reactor during deposition were 240 and 970°C, respectively, the total pressure was 6 mbar, the partial oxygen pressure ${{p}_{{{{{\text{O}}}_{2}}}}}$ was 3 mbar, and the precursor feed rate was 2 mg/min. The deposition was succeeded by annealing in oxygen at a temperature of 970°C for 20 min.

Fig. 1.

(Color online) Setup for preparing thin films by the metal-organic chemical vapor deposition method.

According to the X-ray spectral data, the Fe/Lu ratio in the film obtained was 1.7(3), a value close to $5{\text{/}}3 \approx 1.67$ corresponding to Lu₃Fe₅O₁₂ garnet. The phase composition and orientation of the films obtained were determined from the X-ray θ–2θ scan data. The θ–2θ scan analysis was performed on a Rigaku SmartLab diffractometer (Cu Kα radiation, secondary graphite monochromator) in the angle range of 5°–80° with a step of 0.02° and a signal acquisition time of 1 s. The phase analysis was performed using the JCPDS database. According to the X-ray diffraction data (see Fig. 2), the film exhibits peaks from the Lu₃Fe₅O₁₂ phase and α-Fe₂O₃ impurity (concentration of about 0.3%). The presence of the only (444) reflection from the lutetium iron garnet phase confirms unambiguously the oriented growth of Lu₃Fe₅O₁₂ on the Gd₃Ga₅O₁₂ (111) substrate. The calculated lattice parameter for Lu₃Fe₅O₁₂ is 12.25(1) Å, which is somewhat smaller than the theoretical value for the Lu₃Fe₅O₁₂ phase (12.284 Å) due to elastic deformation of the growing film on the Gd₃Ga₅O₁₂ (111) substrate in view of the lattice mismatch (a(G-d₃Ga₅O₁₂) = 12.383 Å). Thus, the films obtained can be used in experiments on measuring FMR spectra; for spin pumping investigations, a thin (about 10 nm) platinum layer was deposited on the Lu₃Fe₅O₁₂ film by magnetron sputtering.

Fig. 2.

(Color online) θ–2θ scan X-ray diffraction pattern for the Lu₃Fe₅O₁₂ film on the Gd₃Ga₅O₁₂ (111) substrate.

The FMR absorption spectra of the Lu₃Fe₅O₁₂/Pt heterostructure were studied using a system based on a vector network analyzer (VNA); its schematic is shown in Fig. 3. The microwave signal generated by the first VNA port was fed to the first port of a coplanar waveguide. The second VNA port measuring the power of the transmitted microwave signal was connected to the second port of the coplanar waveguide. The coplanar waveguide was placed between the poles of a planar-field electromagnet. The dc magnetic field strength H₀ generated by the electromagnet was perpendicular to the ac magnetic field strength h_ac generated by the coplanar waveguide. The imaginary and real parts of the S₂₁ parameter of the Lu₃Fe₅O₁₂/Pt heterostructure located on the coplanar waveguide were measured. The S₂₁ parameter characterizes the ratio of the power transmitted through the coplanar waveguide to the power at its input. When the FMR frequency and the frequency of the microwave signal fed to the input of the coplanar waveguide coincide, the microwave signal power is absorbed, which can be clearly seen from the S₂₁ parameter. Figure 4 shows the frequency dependence of FMR spectra on the external static magnetic field magnitude H₀. The real and imaginary parts of the S₂₁ parameters of the FMR spectra at different frequencies are given in the insets of Fig. 4. The FMR spectra obtained clearly demonstrate that the resonant field depends on the frequency.

Fig. 3.

(Color online) Schematic of the experimental setup. The ferromagnetic resonance measurement circuit is shown in red. The circuit for measuring the voltage formed in the platinum layer is shown in blue.

Fig. 4.

(Color online) (Circles) Magnetic field dependence of the resonant frequency obtained from the measured ferromagnetic resonance spectra and (pink line) its approximation by the Kittel formula. The magnetic field dependences of the imaginary and real parts of the S₂₁ parameters for different frequencies (with a constant offset along the ordinate axis) are shown in the insets.

When a microwave signal is applied to the heterostructure, uniform precession of the magnetic moment is excited in lutetium garnet. Spin pumping generates the spin current in the ferromagnetic layer. The voltage caused by the inverse spin Hall effect (ISHE voltage) was experimentally studied. It arises in the platinum layer of the Lu₃Fe₅O₁₂/Pt heterostructure because of the conversion of the spin current to the charge current due to the strong spin–orbit interaction in platinum. This voltage was measured using the lock-in detection method (Fig. 3). An amplitude-modulated signal was fed from the radio-frequency generator to the first port of the coplanar waveguide. The second port of the coplanar waveguide was connected to a matched load of 50 Ω. As in the case of measuring FMR spectra, the coplanar waveguide was placed between the poles of the planar-field electromagnet. The Lu₃Fe₅O₁₂/Pt heterostructure was located on the coplanar waveguide, and the waveguide was isolated from the heterostructure because of the conducting platinum layer. Two contacts existing on the coplanar waveguide board are not connected to the radio-frequency line and are used as outputs of the ISHE voltage arising in the platinum layer. These contacts and the platinum layer are connected by silver-based conducting glue. The ISHE voltage was measured using a lock-in amplifier (LIA). A reference signal (which is in phase with the modulating signal) was fed to the LIA from the low-frequency generator output. The voltage from the contacts on the coplanar waveguide was fed to the LIA measuring input. The modulation frequency of the microwave signal was 9.777 kHz. Figure 5 shows the measured ISHE voltage plotted with a constant offset along the ordinate axis at different microwave frequencies. As in the case of FMR, peaks of the measured voltage are shifted to higher resonant fields with an increase in the microwave frequency. When the direction of the external magnetic field changes to opposite, a change in the sign of the measured voltage can be seen, which confirms the origin of this voltage.

Fig. 5.

(Color online) Magnetic field dependences of the measured inverse spin Hall effect voltage for different frequencies with a constant offset along the ordinate axis. Approximation of (red circles) the magnetic field dependence of the measured inverse spin Hall effect voltage by (black line) the theoretical curve at a frequency of 5 GHz is shown in the inset.

The theoretical analysis was performed using the CGS units. It was shown in [8, 21] that the current density caused by spin pumping is proportional to the vector product ${\mathbf{j}} \propto {\mathbf{m}} \times d{\mathbf{m}}{\text{/}}dt$. The j value can be found by solving the Landau–Lifshitz equation with respect to the magnetization vector m divided by the saturation magnetization M_s [4, 22]. This equation can be written as

$$\frac{{d{\mathbf{m}}}}{{dt}} = - \gamma {\mathbf{m}} \times {{{\mathbf{H}}}_{{{\text{eff}}}}} - \alpha \gamma {\mathbf{m}} \times \left[ {{\mathbf{m}} \times {{{\mathbf{H}}}_{{{\text{eff}}}}}} \right],$$

(1)

where γ is the gyromagnetic ratio, α is the damping parameter, and H_eff is the effective magnetic field. The influence of the spin current arising at the interface between a ferromagnet and a heavy metal can be taken into account in (1) as additional damping [8, 23]. In this case, the parameter α in Eq. (1) should be replaced by the effective damping parameter α_eff = ${{\alpha }_{{\text{G}}}} + {{\alpha }_{{{\text{SP}}}}}$, which is the sum of the Gilbert constant α_G [24, 25] and some additive α_SP taking into account the spin current. The effective magnetic field is determined as a variational derivative of the magnetic energy E:

$${{{\mathbf{H}}}_{{{\text{eff}}}}} = - \frac{1}{{{{M}_{s}}}}\frac{{\delta E}}{{\delta {\mathbf{m}}}}.$$

(2)

In the thin-film approximation, the volume magnetic energy density can be written as

$${{E}_{V}} = - {{M}_{s}}\left( {{{{\mathbf{H}}}_{0}} + {{{\mathbf{h}}}_{{{\text{ac}}}}}} \right) \cdot {\mathbf{m}} + \frac{{{{M}_{s}}}}{2}{{M}_{{{\text{eff}}}}}{{\left( {{\mathbf{m}} \cdot {{{\mathbf{e}}}_{z}}} \right)}^{2}}.$$

(3)

Here, ${{{\mathbf{H}}}_{0}} = \left( {0,{{H}_{{0y}}},0} \right)$ is the external static magnetic field, ${{{\mathbf{h}}}_{{{\text{ac}}}}} = \left( {{{h}_{0}}{\text{cos}}(\Omega t),0,0} \right)$ is the ac microwave magnetic field with the frequency Ω and the amplitude ${{h}_{0}} > 0$, and M_eff = 4πM_s – H_p is the effective magnetization, where H_p is the perpendicular magnetic anisotropy. The second term in Eq. (3) describes the shape anisotropy and the perpendicular magnetic anisotropy H_p [26]. We introduce a small vector ${\mathbf{s}}(t)$ describing the deviation of the magnetization vector from the ground state m₀ (note that $\left( {{{{\mathbf{m}}}_{0}} \cdot {\mathbf{s}}} \right) = 0$). Then,

$${\mathbf{m}}(t) = {{{\mathbf{m}}}_{0}} + {\mathbf{s}}(t).$$

(4)

Since ${{{\mathbf{H}}}_{0}} = \left( {0,{{H}_{{0y}}},0} \right)$, ${{{\mathbf{m}}}_{0}} = \left( {0,{{m}_{{0y}}},0} \right)$ and s(t) = $({{s}_{x}}(t),0,{{s}_{z}}(t))$. For definiteness, we assume that ${{H}_{{0y}}} = {{H}_{0}} > 0$ and ${{m}_{{0y}}} \approx 1$. Substituting Eq. (4) into Eq. (1) and retaining the terms up to the first order in s, we obtain the linearized Landau–Lifshitz equation

$$\begin{gathered} \left( {\begin{array}{*{20}{c}} {\frac{{d{{s}_{x}}}}{{dt}}} \\ {\frac{{d{{s}_{z}}}}{{dt}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - {{\alpha }_{{{\text{eff}}}}}\gamma {{H}_{0}}}&{\gamma \left( {{{H}_{0}} + {{M}_{{{\text{eff}}}}}} \right)} \\ { - \gamma {{H}_{0}}}&{ - {{\alpha }_{{{\text{eff}}}}}\gamma \left( {{{H}_{0}} + {{M}_{{{\text{eff}}}}}} \right)} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{s}_{x}}} \\ {{{s}_{z}}} \end{array}} \right) \\ + \left( {\begin{array}{*{20}{c}} {{{\alpha }_{{{\text{eff}}}}}\gamma {{h}_{0}}\cos (\Omega t)} \\ {\gamma {{h}_{0}}\cos (\Omega t)} \end{array}} \right). \\ \end{gathered} $$

(5)

Solutions of Eq. (5) can be sought in the form

$$\begin{array}{*{20}{r}} {{{s}_{x}}}& = &{s_{1}^{x}\cos (\Omega t) + s_{2}^{x}\sin (\Omega t),} \\ {{{s}_{z}}}& = &{s_{1}^{z}\cos (\Omega t) + s_{2}^{z}\sin (\Omega t).} \end{array}$$

(6)

Substituting Eq. (6) into Eq. (5), we can find expressions for the amplitudes $s_{1}^{x},s_{2}^{x},s_{1}^{z},s_{2}^{z}$. Using Eq. (6), we obtain

$$\begin{array}{*{20}{r}} {\mathbf{j}}& = &{{{\kappa }_{j}}\left( {\begin{array}{*{20}{c}} {\Omega (s_{2}^{z}\cos (\Omega t) - s_{1}^{z}\sin (\Omega t))} \\ {\Omega (s_{1}^{z}s_{2}^{x} - s_{1}^{x}s_{2}^{z})} \\ {\Omega (s_{2}^{z}\cos (\Omega t) - s_{1}^{z}\sin (\Omega t))} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{{j}_{x}}} \\ {{{j}_{y}}} \\ {{{j}_{z}}} \end{array}} \right).} \end{array}$$

(7)

Here, ${{\kappa }_{j}}$ is the constant of proportionality between j and ${\mathbf{m}} \times d{\mathbf{m}}{\text{/}}dt$. The components ${{j}_{x}}$ and ${{j}_{z}}$ characterize the response of the heterostructure to the microwave signal at the first harmonic, while the component ${{j}_{y}}$, at the zeroth harmonic. Therefore, the contacts for recording the dc voltage caused by the inverse spin Hall effect lied in the plane perpendicular to the OY axis (Fig. 3). In this case, the recorded voltage V is proportional to ${{j}_{y}} = \Omega (s_{1}^{z}s_{2}^{x} - s_{1}^{x}s_{2}^{z})$. Let us introduce the constant of proportionality $\varkappa$ > 0 between V and j_y. We can write

$$V = \varkappa {{\kappa }_{j}}\frac{{{{\gamma }^{3}}{{\Omega }^{2}}(\alpha _{{{\text{eff}}}}^{2} + 1)({{H}_{0}} + {{M}_{{{\text{eff}}}}})}}{{{{\Omega }^{4}} + {{C}_{1}}{{\Omega }^{2}} + {{C}_{0}}}}h_{0}^{2},$$

(8)

$${{C}_{0}} = {{\gamma }^{4}}H_{0}^{2}{{(\alpha _{{{\text{eff}}}}^{2} + 1)}^{2}}{{({{H}_{0}} + {{M}_{{{\text{eff}}}}})}^{2}},$$

(9)

$${{C}_{1}} = 2{{\gamma }^{2}}(1 - \alpha _{{{\text{eff}}}}^{2})\left( {\frac{{M_{{{\text{eff}}}}^{2}\alpha _{{{\text{eff}}}}^{2}}}{{2(1 - \alpha _{{{\text{eff}}}}^{2})}} + {{H}_{0}}{{M}_{{{\text{eff}}}}} - H_{0}^{2}} \right).$$

(10)

Note that, if Eq. (8) is derived under the assumption that ${{H}_{{0y}}} < 0$ and ${{m}_{{0y}}} \approx - 1$, the ISHE voltage changes the sign, which corresponds to the experimental results (Fig. 5). To find the resonant frequency Ω_r, it is sufficient to differentiate Eq. (8) with respect to Ω and to take into account that the function V(Ω) reaches an extremum at Ω = Ω_r. We obtain the expression

$${{\Omega }_{{\text{r}}}} = \gamma \sqrt {{{H}_{0}}(1 + \alpha _{{{\text{eff}}}}^{2})({{H}_{0}} + {{M}_{{{\text{eff}}}}})} .$$

(11)

If the damping coefficient in (11) is set to zero (${{\alpha }_{{{\text{eff}}}}} = 0$), we obtain the Kittel formula for FMR in a thin magnetic film [27]. Solving Eq. (11) with respect to the external magnetic field, we obtain the resonant field H_r in the form

$${{H}_{{\text{r}}}} = - \frac{1}{2}{{M}_{{{\text{eff}}}}} + \frac{1}{2}\sqrt {M_{{{\text{eff}}}}^{2} + \frac{{4{{\Omega }^{2}}}}{{{{\gamma }^{2}}(1 + \alpha _{{{\text{eff}}}}^{2})}}} .$$

(12)

The substitution of Eq. (11) into Eq. (8) gives the following expression for the resonant ISHE voltage:

$${{V}_{{\text{r}}}}\, = \,\frac{{\varkappa {{\kappa }_{j}}\gamma (1\, + \,\alpha _{{{\text{eff}}}}^{2})\left( {{{M}_{{{\text{eff}}}}}\, + \,\sqrt {M_{{{\text{eff}}}}^{2} + \frac{{4\Omega _{{\text{r}}}^{2}}}{{{{\gamma }^{2}}(1 + \alpha _{{{\text{eff}}}}^{2})}}} } \right)}}{{2{{\alpha }_{{{\text{eff}}}}}\left( {M_{{{\text{eff}}}}^{2} + \frac{{4\Omega _{{\text{r}}}^{2}}}{{{{\gamma }^{2}}(1 + \alpha _{{{\text{eff}}}}^{2})}}} \right)}}h_{0}^{2}.$$

(13)

The gyromagnetic ratio and the effective magnetization can be determined by approximating the dependence of the resonant frequency on the H₀ value (obtained in an FMR experiment, see Fig. 4) using Eq. (11) with ${{\alpha }_{{{\text{eff}}}}} = 0$ (Fig. 1). The found gyromagnetic ratio and effective magnetization are γ = 17.7 MHz/Oe and M_eff = 1223 G. Approximation of the experimental dependence of the ISHE voltage on the external magnetic field for different frequencies made it possible to determine the damping parameter α_eff = $0.089 \pm 0.007$. To exclude $\varkappa$, the coefficient α_eff was chosen on a dimensionless scale (see an example in the inset of Fig. 5). The coefficient $\varkappa$ can be determined by comparing the dependence $V\left( {{{H}_{0}}} \right)$ plotted using Eq. (8) with the experimental results without the introduction of dimensionless variables.

Let us obtain the expression for the sensitivity $K = dV{\text{/}}dP$, where P is the input microwave power. To this end, the relation between the power and the amplitude of the microwave field $P = cQh_{0}^{2}{\text{/}}2$ (c is the speed of light and Q is the sample area) should be taken into account in Eq. (13). Then, the following expression can be written:

$${{V}_{{\text{r}}}} = KP,$$

(14)

where

$$K = \frac{{\varkappa {{\kappa }_{j}}\gamma (1 + \alpha _{{{\text{eff}}}}^{2})\left( {{{M}_{{{\text{eff}}}}} + \sqrt {M_{{{\text{eff}}}}^{2} + \frac{{4\Omega _{{\text{r}}}^{2}}}{{{{\gamma }^{2}}(1 + \alpha _{{{\text{eff}}}}^{2})}}} } \right)}}{{{{\alpha }_{{{\text{eff}}}}}cQ\left( {M_{{{\text{eff}}}}^{2} + \frac{{4\Omega _{{\text{r}}}^{2}}}{{{{\gamma }^{2}}(1 + \alpha _{{{\text{eff}}}}^{2})}}} \right)}}.$$

(15)

Figure 6 shows the dependence of the ISHE voltage on the external static magnetic field H₀ at different real microwave powers P. The structure sensitivity, i.e., the dependence of the rectified voltage on the real power exerted on the sample during measurements, is shown in the inset of Fig. 6. The sensitivity is the slope of V_r(P). Approximation of the experimental results by Eq. (14) yielded K = 8.2 µV/W.

Fig. 6.

(Color online) Magnetic field dependences of the measured inverse spin Hall effect voltage for different microwave powers at a frequency of 10 GHz. The solid line in the inset shows the theoretical dependence of the inverse spin Hall effect voltage on the microwave power, while red circles are the experimental results.

To summarize, it has been demonstrated both theoretically and experimentally that Lu₃Fe₅O₁₂ can be used as a sensitive element for detecting a linearly polarized radio-frequency wave in the range of several tens of gigahertz. Ferromagnetic resonance spectra of the structure have been studied. It has been shown that the theoretical and experimental dependences of the resonant frequency on the magnetic field are in agreement. In addition, the inverse spin Hall effect voltages on the Lu₃Fe₅O₁₂/Pt heterostructure have been measured at different microwave frequencies. The experimental data obtained have been reproduced within the used theoretical model. It has been shown that the resonant frequency increases with the external magnetic field, which is in agreement with theoretical dependence (11). The magnetic field dependence of the inverse spin Hall effect voltage rectified by the Lu₃Fe₅O₁₂/Pt heterostructure shows that the resonance peak amplitude increases with the fed radio-frequency power. Approximation of the experimental data by the theoretical dependences yielded the f-ollowing Lu₃Fe₅O₁₂/Pt parameters: sensitivity 8.2 µV/W, gyromagnetic ratio 17.7 MHz/Oe, and effective magnetization 1223 G. Further improvements of the technology of preparing lutetium garnet films such as the elimination of minor hematite impurities and fitting of the optimal parameters of chemical deposition would facilitate the formation of samples with a narrow ferromagnetic resonance line and a stronger response of the spin pumping of the heterostructure under study. These improvements will increase the sensitivity, thus making it possible to use the Lu₃Fe₅O₁₂/Pt heterostructure in practice as a sensitive element for microwave detectors with the possibility of selective frequency tuning using a static magnetic field.