Dzyaloshinskii-Moriya interactions and adiabatic magnetization dynamics in molecular magnets

A microscopic model of the molecular magnet ${\mathrm{V}}_{15}$ is used to study mechanisms for the adiabatic change of the magnetization in time-dependent magnetic fields. The effects of the Dzyaloshinskii-Moriya interaction, the most plausible source for the energy-level repulsions that lead to adiabatic changes of the magnetization, are studied in detail. We find that the energy-level repulsions that result from this interaction exhibit a strong dependence on the direction of the applied field. We also discuss the role of magnetic anisotropy in the molecule ${\mathrm{Mn}}_{12}-$acetate.

Figure 1

(Color online) Energy levels of Hamiltonian (1) for $J=-2.5\mathrm{K}$, $D_x=D_y=D_z=0.25\mathrm{K}$. At each value of $h$ the slope of each level gives the corresponding value of the projection of the total magnetization on the magnetic field axis. Top: Applied magnetic field $\mathbf{h}$ parallel to the $z$ axis. Bottom: Applied magnetic field $\mathbf{h}$ along the $x$ axis.

Figure 2

(Color online) Top: Energy levels of Hamiltonian (1) for $J=-2.5\mathrm{K}$, $D_x=D_y=D_z=0.25\mathrm{K}$, and the applied magnetic field $\mathbf{h}=h(1,0,1)∕\sqrt{2}$ tilted by 45° with respect to the $z$ axis. Bottom: Detailed view of the $h$ dependence of the four lowest energy levels. At each value of $h$ the slope of each level gives the corresponding value of the projection of the total magnetization on the magnetic field axis.

Figure 3

Schematic diagram of the magnetic interactions in model (1) of the ${\mathrm{V}}_{15}$ molecule.

Figure 4

(Color online) Top: The eight lowest energy levels of ${\mathrm{V}}_{15}$ model (1) with model parameters taken from Ref. 29 as a function of the applied magnetic field $\mathbf{h}$ parallel to the $z$ axis. The values of the DM vector are given in the text. Bottom: Detailed view of the four lowest energy levels at $h\approx 0$.

Figure 5

(Color online) Same as the right panel in Fig. 4 except that the applied magnetic field $\mathbf{h}=h(1,0,1)∕\sqrt{2}$ is tilted by 45° with respect to the $z$ axis (top) and $\mathbf{h}$ is along the $x$ axis (bottom).

Figure 6

(Color online) Energy-level diagram of the Hamiltonian (1) of a distorted triangle with model parameters $J_{1,2}=-2.5\mathrm{K}$, $J_{2,3}=-2.0\mathrm{K}$, $J_{3,1}=-3.0\mathrm{K}$, and $D_x=D_y=D_z=25\mathrm{K}$. The applied magnetic field $\mathbf{h}=h(1,0,1)∕\sqrt{2}$ is tilted by 45° with respect to the $z$ axis.

Figure 7

(Color online) Same as Fig. 6 except that the applied magnetic field $\mathbf{h}$ is parallel to the $z$ axis (top) and along the $x$ axis (bottom).

Figure 8

Top: Schematic diagram of the magnetic interactions of the simplified model (11) of the ${\mathrm{Mn}}_{12}$ molecule (Ref. 37). Black circles, $S=1∕2$; open circles, $S=2$. Also shown are the DM vectors (for $i<j$ and ${\mathbf{D}}_{i,j}=-{\mathbf{D}}_{j,i}$). Bottom: (Color online) The 21 lowest energy levels of the ${\mathrm{Mn}}_{12}$ model (11) as a function of the applied magnetic field $\mathbf{h}$. Solid lines, eigenstates with $S\approx 10$; dashed lines, eigenstates with $S\approx 9$. The applied magnetic field $\mathbf{h}$ is parallel to the $z$ axis.