Unquenched orbital angular momentum as the origin of spin inertia

Abstract

The recent proposal and observation of spin inertia, and the consequent high-frequency spin nutation mode, have raised key questions for our understanding of magnetization dynamics, especially considering its high relevance for magnetic memories and ultrafast switching. Notwithstanding recent progress, a clear identification of spin inertia's physical origin thereby offering predictive power remains to be accomplished. Here, discussing general principles for identifying this physical origin, we examine unquenched orbital angular momentum (OAM) finding it to be a key candidate, despite its typically small value. Treating OAM and spin within a two-sublattice model, we derive the equivalent single-sublattice framework for magnetization dynamics making appropriate approximations. The latter naturally manifests the spin inertia term and parameter, which are otherwise introduced phenomenologically. The inertia parameter evaluated within our model is found to be in good agreement with its experimentally observed value in cobalt. We further delineate key experimental signatures that could verify or rule out the unquenched OAM as the origin of the observed high-frequency mode, and avoid a spurious optical mode in a two-sublattice ferromagnet from being identified as nutation. Our analysis offers a potential link between the recently emerged fields of orbitronics and spin inertia, thereby motivating investigations at their intersection.

Figures (4)

• Figure 1: Schematic depiction of the model. (a) Visualization of Russel-Saunders (RS) coupling. Top panel: An electron in black carrying its spin $\va{S}$ in blue orbiting around the atom core in grey with the orbital angular momentum $\va{L}$. These angular momenta couple on the atomic level via RS coupling with the coupling parameter $\lambda$. This repeats at each atomic site. Bottom panel: The RS coupling is distinct from the exchange coupling $J$ between the spins of electrons at different sites $S_i$ and $S_j$. (b) Visualization of nutation from coupled precessions. The spin magnetization $\va{M}_S$ in blue and the orbital magnetization $\va{M}_L$ in red each precess around their effective magnetic field $\va{H}_S$ for $\va{M}_S$ and $\va{H}_L$ for $\va{M}_L$ in green.
• Figure 2: Dependence of the precession and nutation frequencies for ferromagnetic and antiferromagnetic RS coupling on the relative strength of $\lambda$ (a) and $M_{L0}$ (b). For a typical material, $\lambda/K_S$ is estimated at $10^4$ such that the two frequencies are far apart from each other. The parameters are chosen to represent typical material values Laughin2014Chikazumi2009 at $M_{S0}=1.5\ \text{J$\cdot$T$^{-1}\cdot$cm$^{-3}$}$, $M_{L0}/M_{S0}=0.1$, $K_S=K_L=2.5\times10^{-2}\ \text{cm$^3\cdot$T$^2\cdot$J}^{-1}$, $\mu_0H_0/K_SM_{S0}=2$, $\abs{\gamma_L}=1.76\times10^{11}\ \text{s$^{-1}\cdot$T}^{-1}$ and $\abs{\gamma_S}/\abs{\gamma_L}=2$ for (a) and $M_{S0}=1.5\ \text{J$\cdot$T$^{-1}\cdot$cm}^{-3}$, $K_S=K_L=2.5\times10^{-2}\ \text{cm$^3\cdot$T$^2\cdot$J}^{-1}$, $\mu_0H_0/K_SM_{S0}=2$, $\lambda/K_S=40$, $\abs{\gamma_L}=1.76\times10^{11}\ \text{s$^{-1}\cdot$T}^{-1}$ and $\abs{\gamma_S}/\abs{\gamma_L}$=2 for (b). For $\omega_{A2}$ the absolute value is depicted to maintain comparability, as a result there appears to be a crossings of $\omega_{A1}$ and $\omega_{A2}$ but this is only a consequence of the visualization and has no physical meaning.
• Figure 3: Depiction of the simplified magnetization mode profiles using $\frac{\abs{\gamma_S}}{\abs{\gamma_L}}=2$ and $C_F\approx 1$, $C_A\approx 1$ with (a) the ferromagnetic precession mode with $\omega_{F1}$, (b) the ferromagnetic nutation mode with $\omega_{F2}$, (c) the antiferromagnetic precession mode with $\omega_{A1}$ and (d) the ferromagnetic nutation mode with $\omega_{A2}$. $m$ in (b) and (d) is the magnetic amplitude in $\va{M}_L$.
• Figure 4: Dynamic susceptibility $\abs{\chi_{c 1}+\chi_{c 2}}$ for (a) ferromagnetic and (b) antiferromagnetic RS coupling for $\lambda/K_S=10$ and $\lambda/K_S=100$. The other parameters are $K_S=K_L=2.5\times10^{-2}\ \text{cm$^3\cdot$T$^2\cdot$J}^{-1}$, $M_{S0}=1.5\ \text{J$\cdot$T$^{-1}\cdot$cm$^{-3}$}$, $M_{L0}/M_{S0}=0.1$, $\mu_0H_0/K_SM_{S0}=0.2$, $\abs{\gamma_L}=1.76\times10^{11}\ \text{s$^{-1}\cdot$T}^{-1}$, $\abs{\gamma_S}/\abs{\gamma_L}$=2, $\alpha_{SS}/\abs{\gamma_S}M_{S0}K_S=\alpha_{LL}/\abs{\gamma_L}M_{L0}K_S=0.004$ and $\alpha_{SL}/\abs{\gamma_S}M_{L0}K_S=\alpha_{LS}/\abs{\gamma_L}M_{S0}K_S=0.002$.