Spin Pumping into Anisotropic Dirac Electrons

In spintronics, spin currents are crucial in using electrons’ charge and spin. Spin pumping, the spin current generation of conduction electrons from nonequilibrium magnetization dynamics at magnetic interfaces, is a popular method for generating and manipulating spin currents. In previous experimental reports on spin pumping, the enhancement of Gilbert damping in ferromagnetic resonance (FMR) was observed due to the loss of angular momentum associated with the spin current injection into the nonmagnetic layer adjacent to the ferromagnetic layer Heinrich et al. (1987); Celinski and Heinrich (1991); Mizukami et al. (2001a, b, 2002); Ingvarsson et al. (2002); Lubitz et al. (2003); Sarma (2004); Tserkovnyak et al. (2005). Mizukami et al. measured the enhancement of the Gilbert damping associated with the adjacent nonmagnetic metal. They reported that the strong spin-orbit coupling in the nonmagnetic layer strictly affected the enhancement of the Gilbert damping Mizukami et al. (2001a, b, 2002). Consequently, electric detection by inverse spin Hall effect, in which the charge current is converted from the spin current, led to spin pumping being used as an essential technique for studying spin-related phenomena in nonmagnetic materials Azevedo et al. (2005); Saitoh et al. (2006); Ando et al. (2008, 2009, 2011); Mosendz et al. (2010a, b); Czeschka et al. (2011); Miron et al. (2011); Liu et al. (2011, 2012); Kajiwara et al. (2010); Bai et al. (2013); Sandweg et al. (2011); Sinova et al. (2015). Saitoh et al. measured electric voltage in a bilayer of Py and Pt under microwave application. They observed that charge current converted because of inverse spin Hall effect from spin current injected by spin pumping Saitoh et al. (2006).

In the first theoretical report on spin pumping, Berger predicted an increase in Gilbert damping due to the spin current flowing interface between the ferromagnetic and nonmagnetic layers Berger (1996); Hellman et al. (2017). Tserkovnyak et al. calculated the spin current flowing through the interface Tserkovnyak et al. (2002a, b, 2003) based on the scattering-matrix theory and the picture of adiabatic spin pumping Mucciolo et al. (2002); Sharma and Chamon (2003); Watson et al. (2003). They introduced a complex spin-mixing conductance that characterizes spin transport at the interfaces based on spin conservation and no spin loss. The spin mixing conductance can represent the spin pumping-associated phenomena and is quantitatively evaluated using the first principle calculation Xia et al. (2002). Nevertheless, microscopic analysis is necessary to understand the detailed mechanism of spin transport at the interface Ohnuma et al. (2014); Matsuo et al. (2018); Kato et al. (2019, 2020); Ominato and Matsuo (2020); Ominato et al. (2020); Yamamoto et al. (2021); ominatoAnisotropicSuperconductingSpin2021 ; Yama et al. (2021); Yama2022 ; Ominato et al. (2022). It was clarified that spin pumping depends on the anisotropy of the electron band structure and spin texture. Spin pumping is expected to be one of the probes of the electron states ominatoAnisotropicSuperconductingSpin2021 ; Yama et al. (2021); Yama2022 ; Ominato et al. (2022).

Bismuth has been extensively studied because of its attractive physical properties, such as large diamagnetism, large $g$-factor, high efficient Seebeck effect, Subrikov-de Haas effect, and de Haas-van Alphen effect Édel’man (1976); Fuseya et al. (2015). The electrons in the conduction and valence bands near the $L$-point in bismuth, which contribute mainly to the various physical phenomena, are expressed as effective Dirac electrons. Thus, electrons in bismuth are called Dirac electrons Wolff (1964); Édel’man (1976); Fuseya et al. (2015). The doping antimony to bismuth is known to close the gap and makes it a topological insulator Fu and Kane (2007); Teo et al. (2008). Because of its strong spin-orbit interaction, bismuth has attracted broad attention in spintronics as a high efficient charge-to-spin conversion material Fuseya et al. (2012a, b, 2014); Fukazawa et al. (2017); Yue et al. (2018); Chi et al. (2020). The spin current generation at the interface between the bismuth oxide and metal has been studied since a significant Rashba spin-orbit interaction appears at the interface Karube et al. (2016). The spin injection into bismuth was observed due to spin pumping from yttrium iron garnet or permalloy Hou et al. (2012); Emoto et al. (2014, 2016). Nevertheless, microscopic analysis of spin pumping into bismuth has not been performed. The dependence of the spin pumping on the crystal and band structure of bismuth remains unclear.

This study aims at a microscopic analysis of spin injection due to spin pumping into an anisotropic Dirac electron system, such as bismuth, and investigates the dependence of spin pumping on the band structure. We consider a bilayer system comprising an anisotropic Dirac electron system and a ferromagnetic insulator where a microwave is applied (see Fig. 1). The effect of the interface is treated by proximity exchange coupling between the Dirac electron spins and the localized spins of the ferromagnetic insulator Ohnuma et al. (2014); Matsuo et al. (2018); Kato et al. (2019, 2020); Ominato and Matsuo (2020); Ominato et al. (2020); Yamamoto et al. (2021); ominatoAnisotropicSuperconductingSpin2021 ; Yama et al. (2021); Yama2022 ; Ominato et al. (2022). We calculate the Gilbert damping enhancement due to spin pumping from the ferromagnetic insulator into the Dirac electron system up to the second perturbation of the interfacial exchange coupling. For illustarion, we calculate the enhancement of the Gilbert damping for an anisotropic Dirac system in bismuth.

This paper is organized as follows: Sec. II describes the model. Sec. III shows the formulation of the Gilbert damping enhancement and discuss the effect of the interfacial randomness on spin pumping. Sec. IV summarizes the results and demonstration of the Gilbert damping enhancement in bismuth. Sec. V presents the conclusion. The Appendices show the details of the calculation. Appendix A defines the magnetic moment of electrons in a Dirac electron system. Appendix B provides the detailed formulation of the Gilbert damping modulation, and Appendix C presents the detailed derivation of Gilbert damping modulation.

We consider a bilayer system composed of an anisotropic Dirac electron system and a ferromagnetic insulator under a static magnetic field. We evaluate a microscopic model whose Hamiltonian is given as

where $\hat{\mathcal{H}}_{D}$, $\hat{\mathcal{H}}_{\text{FI}}$, and $\hat{\mathcal{H}}_{\text{ex}}$ represent an anisotropic Dirac electron system, a ferromagnetic insulator, and an interfacial exchange interaction, respectively.

Applying a microwave to the ferromagnetic insulator includes the localized spin’s precession. The Gilbert damping constant can be read from the retarded magnon Green function defined by

with $S^{+}_{\bm{k}}(t)=e^{i\hat{\mathcal{H}}_{T}/\hbar}S^{+}_{\bm{k}}e^{-i\hat{\mathcal{H}}_{T}/\hbar}$ being the Heisenberg representation of the localized spin, since one can prove that the absorption rate of the microwave is proportional to ${\rm Im}\,G_{{\bm{k}}={\bm{0}}}^{R}(\omega)$ (see also Appendix B). By considering the second-order perturbation with respect to the matrix element for the spin transfer $\mathcal{T}_{\bm{q},\bm{k}}$, the magnon Green function is given by Ohnuma et al. (2014); Matsuo et al. (2018); Kato et al. (2019, 2020); Ominato and Matsuo (2020); Ominato et al. (2020); Yamamoto et al. (2021); ominatoAnisotropicSuperconductingSpin2021 ; Yama et al. (2021); Yama2022 ; Ominato et al. (2022)

Here, we introduced a term, $i\alpha\omega$, in the denominator to express the spin relaxation within a bulk FI, where $\alpha$ indicates the strength of the Gilbert damping. The enhancement of the damping, $\delta\alpha$, is due to the adjacent Dirac electron system, calculated by

where $\chi^{R}_{\bm{q}}(\omega)$ is the retarded component of the spin susceptibility (defined below). We assume that the FMR peak described by ${\rm Im}\,G_{{\bm{k}}={\bm{0}}}^{R}(\omega)$ is sufficiently sharp, i.e., $\alpha+\delta\alpha\ll 1$. Then, the enhancement of the Gilbert damping can be regarded as almost constant around the peak ($\omega\simeq\omega_{0}$), allowing us to replace $\omega$ in $\delta\alpha$ with $\omega_{\bm{0}}$.

The retarded component of the spin susceptibility for the Dirac electrons:

The retarded component of the spin susceptibility is derived from the following Matsubara Green function through analytic continuation $i\omega_{l}\to\hbar\omega+i\delta$:

where $\omega_{l}=2\pi l/\beta$ is the bosonic Matsubara frequency with $l$ being integers. According to Wick’s theorem, the Matsubara representation of the spin susceptibility is given by

where $\sum_{i\epsilon_{n}}$ indicates the sum with respect to the fermionic Matsubara frequency, $\epsilon_{n}=(2\pi+1)n/\beta$. The imaginary part of the spin susceptibility is given by

where $f(\epsilon)=(e^{\beta(\epsilon-\mu)}+1)^{-1}$ is the Fermi distribution function, $\lambda=\pm$ is a band index (see Fig. 3), and $\mathcal{F}(\theta,\phi)$ is the dimensionless function which depends on the direction of the ordered localized spin, defined by

For detailed derivation, see Appendix C.

In this paper, we model the interfacial spin transfer as a combination of the clean and dirty processes. The former corresponds to the momentum-conserved spin transfer and the latter to the momentum-nonconserved one ominatoAnisotropicSuperconductingSpin2021 ; Ominato et al. (2022). By averaging over the position of the localized spin at the interface, we can derive the matrix elements of the interfacial spin-transfer process as

where $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are the averaged matrix elements contributing to the clean and dirty processes, respectively. Then, the enhancement of the Gilbert damping is given by

where $\chi^{R}_{\text{uni}}(\omega)$ and $\chi^{R}_{\text{uni}}(\omega)$ are the local and uniform spin susceptibilities defined by

respectively. From Eq. (III), their imaginary parts are calculated as

The enhancement of the Gilbert damping, $\delta\alpha$, depends on the direction of the ordered localized spin through the dimensionless function $\mathcal{F}(\theta,\phi)$ regardless of the interfacial condition.

By contrast, the FMR frequency dependence of $\delta\alpha$ reflects the interfacial condition; for a clean interface, it is determined mainly by $\text{Im}\,\chi^{R}_{\text{uni}}(\omega_{{\bm{0}}})$, whereas for a dirty interface, it is determined by $\text{Im}\,\chi^{R}_{\text{loc}}(\omega_{{\bm{0}}})$. The FMR frequency dependence of the local and uniform spin susceptibilities, $\text{Im}\,\chi^{R}_{\text{loc}}(\omega_{{\bm{0}}})$ and $\text{Im}\,\chi^{R}_{\text{uni}}(\omega_{{\bm{0}}})$, are plotted in Figs. 4 (a) and (b), respectively. The local and uniform spin susceptibilities are normalized by $\pi n_{D}^{2}\nu_{0}^{2}$ and $\pi n_{D}\nu_{0}$, respectively, where $\nu_{0}\equiv 1/2\pi^{2}\hbar^{3}\sqrt{\text{det}\alpha_{ij}}$ is defined. In the calculation, the ratio of the chemical potential to the energy gap was set to $\mu/\Delta\simeq 4.61$, which is the value in the bismuth Fuseya et al. (2015). According to Fig. 4 (a), the local spin susceptibility increases linearly with the frequency $\omega$ in the low-frequency region. This $\omega$-linear behavior can be reproduced analytically for low temperatures and $\hbar\omega\ll\mu$:

Fig. 4 (b) indicates a strong suppression of the uniform spin susceptibility below a spin-excitation gap ($\omega_{\bm{0}}<2\mu$). This feature can be checked by its analytic form at zero temperature:

Thus, the FMR frequency dependence of the enhancement of the Gilbert damping depends on the interfacial condition. This indicates that the measurement of the FMR frequency dependence may provide helpful information on the randomness of the junction.

We consider bismuth, which is one of the anisotropic Dirac electron systems Édel’man (1976); Tang and S. Dresselhaus (2014); Fuseya et al. (2014, 2015); Zhu et al. (2011). The crystalline structure of pure bismuth is a rhombohedral lattice with the space group of $R\bar{3}m$ symmetry, see Figs. 5 (a) and (b). It is reasonable to determine the Cartesian coordinate system in the rhombohedral structure using the trigonal axis with $C_{3}$ symmetry, the binary axis with $C_{2}$ symmetry, and the bisectrix axis, which is perpendicular to the trigonal and binary axes. Hereafter, we choose the $x$-axis as the binary axis, the $y$-axis as the bisectrix axis, and the $z$-axis as the trigonal axis. Note that the trigonal, binary, and bisectrix axes are denoted as $[0001]$, $[1\bar{2}10]$, and $[10\bar{1}0]$, respectively, where the Miller-Bravais indices are used. The bismuth’s band structure around the Fermi surface consists of three electron ellipsoids at $L$-points and one hole ellipsoid at the $T$-point. It is well known that the electron ellipsoids are the dominant contribution to the transport phenomena since electron’s mass is much smaller than that of the hole, see Fig. 5 (c). Therefore, the present study considers only the electron systems at the $L$-points. The electron ellipsoids are significantly elongated, with the ratio of the major to minor axes being approximately $15:1$. Each of the three electron ellipsoids can be converted to one another with $2\pi/3$ rotation around the trigonal axis. The electron ellipsoid along the bisectrix axis is labeled as $e1$, and the other two-electron ellipsoids are labeled $e2$ and $e3$. The inverse mass tensor for the $e1$ electron ellipsoids is given by

The inverse mass tensor of the electron ellipsoids $e2$ and $e3$ are obtained by rotating that of $e1$ by $2\pi/3$ rotation as below:

Let us express the dimensionless function $\mathcal{F}(\theta,\phi)$ representing the localized spin direction dependence of the damping enhancement on the inverse mass tensors.

Here, we use the following calculations:

where $i,j,k$ are cyclic. $(\cdots)_{\text{total}}$ represents the summation of the contributions of the three electron ellipsoids, and $\bar{\kappa}_{\parallel}$, $\bar{\kappa}_{\perp}$ $(>0)$ are the total Gaussian curvature of the three electron ellipsoids normalized by the electron mass $m$, given by

Hence, the dimensionless function $\mathcal{F}$ is given by

The results suggest that the variation of the damping enhancement depends only on the polar angle $\theta$, which is the angle between the direction of the ordered localized spin $\langle\bm{S}\rangle_{0}$ and the trigonal axis. It is also found that the $\theta$ dependence of the damping enhancement originates from the anisotropy of the band structure. The dimensionless function $\mathcal{F}(\theta)$ is plotted in Fig. 6 by varying the ratio of the total Gaussian curvatures $x=\bar{\kappa}_{\perp}/\bar{\kappa}_{\parallel}$, which corresponds to the anisotropy of the band structure. Figure 6 shows that the $\theta$-dependence of the damping enhancement decreases with smaller $x$ and the angular dependence vanishes in an isotropic Dirac electron system $x=1$. Bismuth is known to have a strongly anisotropic band structure. The magnitude of the matrix elements of the inverse mass $\alpha_{1}$-$\alpha_{4}$ was experimentally determined as $m\alpha_{1}=806$, $m\alpha_{2}=7.95$, $m\alpha_{3}=349$, and $m\alpha_{4}=37.6$. The total Gaussian curvatures are evaluated as Fuseya et al. (2015)

The ratio of the total Gaussian curvature is estimated as $x\simeq 22.1$. Therefore, the damping enhancement is expected to depend strongly on the polar angle $\theta$ in a bilayer system composed of single-crystalline bismuth and ferromagnetic insulator. Conversely, the $\theta$-dependence of the damping enhancement is considered to be suppressed for polycrystalline bismuth.

The damping enhancement is independent of the azimuthal angle $\phi$. Therefore, it is invariant even on rotating the spin orientation around the trigonal axis. The reason is that the azimuthal angular dependence of the damping enhancement cancels out when the contributions of the three electron ellipsoids are summed over, although each contribution depends on the azimuthal angle. The azimuthal angular dependence of the damping enhancement is expected to remain when strain breaks the in-plane symmetry. Additionally, suppose the spin can be injected into each electron ellipsoid separately, e.g., by interfacial manipulation of the bismuth atoms. In that case, the damping enhancement depends on the azimuthal angle of the spin orientation of the ferromagnetic insulator Ominato et al. (2020). This may be one of the probes of the electron ellipsoidal selective transport phenomena.

It is also noteworthy that the damping enhancement varies according to the ordered localized spin direction with both clean and dirty interfaces; that is independent of whether momentum is conserved in interfacial spin transport. Conversely, it was reported that the spin orientation dependence of the damping enhancement due to the Rashba and Dresselhaus spin-orbit interaction turned out to vanish by interfacial inhomogeneity Yama et al. (2021); Yama2022 .

We theoretically studied spin pumping from a ferromagnetic insulator to an anisotropic Dirac electron system. We calculated the enhancement of the Gilbert damping in the second perturbation concerning the proximity interfacial exchange interaction by considering the interfacial randomness. For illustration, we calculated the enhancement of the Gilbert damping for an anisotropic Dirac system realized in bismuth. We showed that the Gilbert damping varies according to the polar angle between the ordered spin $\langle\bm{S}\rangle_{0}$ and the trigonal axis of the Dirac electron system whereas it is invariant in its rotation around the trigonal axis. Our results indicate that the spin pumping experiment can provide helpful information on the anisotropic band structure of the Dirac electron system.

The Gilbert damping is invariant in the rotation around the trigonal axis because the contributions of each electron ellipsoid depend on the in-plane direction of the ordered spin $\langle\bm{S}\rangle_{0}$. Nevertheless, the total contribution becomes independent of the rotation of the trigonal axis after summing up the contributions from the three electron ellipsoids that are related to each other by the $C_{3}$ symmetry of the bismuth crystalline structure. If the spin could be injected into each electron ellipsoid separately, it is expected that the in-plane direction of the ordered localized spin would influence the damping enhancement. This may be one of the electron ellipsoid selective spin injection probes. The in-plane direction’s dependence will also appear when a static strain is applied. A detailed discussion of these effects is left as a future problem.