Butterfly hysteresis curves generated by adiabatic Landau-Zener transitions

Butterfly hysteresis curves observed in dynamical magnetization measurements on systems of low-spin magnetic molecules such as ${\mathrm{V}}_{15}$ can be explained by the interplay of adiabatic Landau-Zener transitions and relaxation. We investigate the magnetization dynamics analytically in the basis of the adiabatic energy levels of the spin $1∕2$, to a qualitative accordance with experimental observations. In particular, reversible behavior is found near zero field, the corresponding susceptibility being bounded by the equilibrium and adiabatic susceptibilities.

Figure 1

(Color online) Adiabatic Landau-Zener effect with thermal transitions. The direction of relaxation changes at the energy bias $W_{\mathrm{BC}}$, which corresponds to the butterfly crossing of the magnetization curve, $m_z\left(t\right)=m_z^{\left(\mathrm{eq}\right)}\left(t\right)$.

Figure 2

The population difference $n_{-}\left(t\right)-n_{+}\left(t\right)$ obtained by numerical integration in Eq. (6) vs the energy bias $W$ for the linear sweep from $t=-\infty$ to $\infty$ in the limit $\Delta ∕k_{B}T\to 0$ for different values of the reduced sweep rate $A$ of Eq. (17), $I_{\alpha }$ is given by Eq. (21).

Figure 3

(Color online) Magnetization hysteresis curves $m_z\left(t\right)$ for ${\mathrm{V}}_{15}$.

Figure 4

(Color online) Susceptibility at the level crossing vs $A$ for $\alpha =3$ and different values of $\Delta ∕\left(k_{B}T\right)$. The asymptotes given by Eqs. (26, 27) are shown by the dashed lines.

Figure 5

(Color online) Butterfly-crossing bias $W_{\mathrm{BC}}$ vs the reduced relaxation time $A$ for $\alpha =1.5$. The nearly equilibrium asymptotes of Eq. (22) on the left side and the nearly adiabatic asymptote of Eq. (19) on the right side are shown by the dashed lines.