Transverse susceptibility and resonance absorption of an anti-ferromagnetic spin wave interacting with a phonon reservoir

Abstract

Abstract A form of the transverse magnetic susceptibility for an anti-ferromagnetic spin system interacting with a phonon reservoir is derived employing the TCLE method (a method in which the admittance of a physical system is directly derived from time-convolutionless equations with external driving terms) in terms of the non-equilibrium thermo-field dynamics (NETFD) in the spin-wave approximation. The region valid for the lowest spin-wave approximation is numerically investigated in detail, and the transverse susceptibility for the anti-ferromagnetic system of one-dimensional infinite spins of magnitude $S = 2$ is numerically studied in that region. It is confirmed that the effects of the memory and initial correlation for the spin system and phonon reservoir, which are represented by the interference terms in the TCLE method, increase the power absorption in the resonance region. In the region valid for the lowest spin-wave approximation, it is shown that the line half-width $\Delta \omega_{\mathrm{RF}}$ of the power absorption in the resonance region for a transversely rotating magnetic field increases as the temperature $T$ rises, and decreases as the uniaxial anisotropy energy $\hbar K$ of the $z$ direction increases or as the wave number $k$ becomes large, and also that the line peak-height $H_{\mathrm{RF}}$ of the power absorption in the resonance region for the transverse magnetic field decreases as $T$ rises and increases as $\hbar K$ increases or as $k$ becomes large. According to analytic considerations, it is anticipated that $\Delta \omega_{\mathrm{RF}}$ and $H_{\mathrm{RF}}$ vary approximately as $\Delta \omega_{\mathrm{RF}} \approx \{A_k + B_k \bar{n} (\omega_{{\mathrm{R}} k} )\} \{ \bar{n}(\omega_{{\mathrm{R}} k}) + \,1\}$ and $H_{\mathrm{RF}} \propto \{A_k + B_k \bar{n}(\omega_{{\mathrm{R}} k})\}^{- 1} \{ \bar{n}(\omega_{{\mathrm{R}} k}) + 1\}^{- 1}$, with $\bar{n}(\omega_{{\mathrm{R}} k}) = \{ \exp(\hbar \omega_{{\mathrm{R}} k}/ (k_{\mathrm{B}} T)) -1 \}^{- 1}$, where $\omega_{{\mathrm{R}} k}$ is the characteristic frequency of the phonon reservoir. Here, $A_k$ is a positive quantity dependent on the Zeeman frequency, the spin magnitude, and the anisotropy energy of the spin system, and $B_k$ is a positive quantity dependent on the anisotropy energy. At low temperatures ($k_{\mathrm{B}} T \ll \hbar \omega_{{\mathrm{R}} k}$), it is anticipated that $\Delta \omega_{\mathrm{RF}}$ and $H_{\mathrm{RF}}$ vary approximately as $\Delta \omega_{\mathrm{RF}} \approx A_k \exp\{((A_k + B_k)/ A_k) \exp(- \hbar \omega_{{\mathrm{R}} k}/ (k_{\mathrm{B}} T))\}$ and $H_{\mathrm{RF}} \propto A_k^{- 1} \exp\{- ((A_k + B_k)/ A_k) \exp(- \hbar \omega_{{\mathrm{R}} k}/ (k_{\mathrm{B}} T))\}$.