Frequency-domain magnetic-resonance spectroscopic investigations of the magnetization dynamics in ${\text{Mn}}_{12}\text{Ac}$ single crystals

A comprehensive spectroscopic study of the magnetization relaxation of ${\text{Mn}}_{12}\text{Ac}$ is presented. The single crystalline samples are investigated as a function of magnetic field and temperature using frequency-domain magnetic-resonance spectroscopy. The magnetization relaxation is followed in real-time by recording spectra as a function of delay time. The experiments are performed both in the absence of a magnetic field and in external fields either parallel (Faraday geometry) or perpendicular (Voigt geometry) to the direction of radiation. The relaxation rates obtained from spectral fits correspond well to those obtained from magnetometric measurements. The results exhibit clear signatures of quantum tunneling of the magnetization. The influence of $D$ strain and internal dipolar fields is observed.

Figure 1

(a) Schematic spectrometer setup showing the directions of the external magnetic field $\mathbf{H}$, the radiation propagation direction $\mathbf{q}$, and the magnetic field direction of the terahertz radiation $\mathbf{h}$. In Faraday geometry $\mathbf{h}\perp \mathbf{H}$, while in Voigt geometry $\mathbf{h}\perp \mathbf{H}$ or $\mathbf{h}\parallel \mathbf{H}$. (b) Energy levels in the presence of an external magnetic field $H>0$. (c) Arrangement of the energy levels after inversion of the magnetic field.

Figure 2

(Color online) Zero-field FDMRS spectra recorded on sample V of ${\text{Mn}}_{12}\text{Ac}$ at (a) $T=5\text{K}$ and $\mathbf{h}\parallel \mathbf{c}$, and (b) for various temperatures and $\mathbf{h}\perp \mathbf{c}$, with the crystal $c$ axes in the plane of the 0.49 mm thick sample.

Figure 3

(a) Temperature dependence of the resonance-line intensities $\Delta \mu$ obtained from the fits of the transmission spectra for the $|\pm 10⟩\to |\pm 9⟩$ and $|\pm 9⟩\to |\pm 8⟩$ transitions. The lines correspond to theoretically predicted temperature dependences of $\Delta \mu$ taken into account the ground-state multiplet only (dotted lines) or excited states, too (solid line). (b) Far-infrared transmission spectra recorded on a pressed powder pellet of ${\text{Mn}}_{12}\text{Ac}$ at $H=0$ and $H=7\text{T}$. Two magnetic-resonance lines are observed at low frequencies and two optical phonon transitions at 35 and $53{\text{cm}}^{-1}$, respectively. Panel C shows the ratio of the transmission spectra: $\text{Tr}\left(H=7\text{T}\right)$ divided by the zero-field data.

Figure 4

(Color online) (a) FDMRS spectra of ${\text{Mn}}_{12}\text{Ac}$ recorded at on (field-cooled) sample V for different static magnetic fields $\mathbf{H}\parallel \mathbf{c}$, where the temperature was $T=2.35\text{K}$ $\left(H<2\text{T}\right)$ and 3.0 K $\left(H\ge 2\text{T}\right)$. The magnetic absorption line shifts monotonously to higher frequencies as the applied field $\mathbf{H}$ increases. The field strength applied is (from left to right): 0, 0.45, 0.90, 1.35, 1.80, 2.25, 2.50, 3.00, 4.00, 6.00, and 7.00 T). (b) Resonance frequencies as a function of external magnetic field $\left(\mathbf{H}\parallel \mathbf{c}\right)$ obtained from measurements on sample V at $T=14\text{K}$ (symbols). The drawn lines were calculated according to ${\nu }_{\text{res}}={\nu }_0+g_z{\mu }_B{H}_z$ with $g=1.94$. (c) Gaussian linewidth as a function of magnetic field.

Figure 5

(Color online) FDMRS spectra recorded on sample V of ${\text{Mn}}_{12}$ at $T=1.75\text{K}$ and various delay times in minutes as indicated. (a) The absorption line corresponding to the $|+10⟩\to |+9⟩$ transition; (b) the resonance line of the $|-10⟩\to |-9⟩$ transition. The solid lines correspond to the calculations from which the population of the respective states can be obtained. (c) Relative populations of the metastable $|-10⟩$ state (${\rho }_{-10}$, open dots) and the ground $|+10⟩$ state (${\rho }_{+10}$, full dots).

Figure 6

(Color online) Dependence of the relaxation time on the longitudinal magnetic field. The data are extracted from Voigt geometry measurements $\left(T=1.77\text{K}\right)$ of the resonance line due to the transition from the metastable state to the ground state (open green circles) and due to the transition from the ground state to the metastable state (closed blue circles). In addition data from measurement in Faraday geometry at 1.86 K are plotted (brown triangles) obtained for fits of the absorption line due to the transition from the ground state to the metastable state. The solid line represents the calculated relaxation time from the phonon-assisted spin tunneling combined with the thermal activation process.

Figure 7

(Color online) FDMRS spectra recorded on sample V of ${\text{Mn}}_{12}$ at $T=1.75\text{K}$ and various delay times. The data in panel A are recorded at $H=1.80\text{T}$, the data in panel B at $H=1.93\text{T}$. Note, that for clarity reasons only a few of the recorded spectra are displayed here. Panel C gives the frequency shift of the resonance line between the first recorded and last recorded spectra for the $|+10⟩\to |+9⟩$ transition as a function of longitudinal magnetic field.

Figure 8

(a) With increasing internal dipolar field the energy difference of the $|-9⟩$ and $|-10⟩$ states decreases, while the transition frequency of the $|+10⟩\to |+9⟩$ increases. (b) Time-dependent local dipolar field $H_{\text{dip}}\left({\rho }_{+10}-{\rho }_{-10}\right)$ plotted as a function of time. The experimental data belongs to the measurement at $H=1.85\text{T}$, $T=1.77\text{K}$ (see text).

Figure 9

(Color online) FDMRS transmission spectra recorded on sample F at various delay times as indicated. (a) The data were taken at $T=1.86\text{K}$ and $H_z=1.95\text{T}$, and (b) $T=3.3\text{K}$ and $H_z=0\text{T}$. (c) Population of the $|+10⟩$ ground state as a function of time for the measurements in the upper panels.