Master equations for pulsed magnetic fields: Application to magnetic molecules

We extend spin-lattice relaxation theory to incorporate the use of pulsed magnetic fields for probing the hysteresis effects and magnetization steps and plateaus exhibited, at low temperatures, by the dynamical magnetization of magnetic molecules. The main assumption made is that the lattice degrees of freedom equilibrate in times much shorter than both the experimental time scale (determined by the sweep rate) and the typical spin-lattice relaxation time. We first consider the isotropic case (a magnetic molecule with a ground state of spin $S$ well separated from the excited levels and also the general isotropic Heisenberg-Hamiltonian where all energy levels are relevant) and then we include small off-diagonal terms in the spin Hamiltonian to take into account the Landau-Zener-Stückelberg (LZS) effect. In the first case, and for an $S=1∕2$ magnetic molecule we arrive at the generalized Bloch equation recently used for the magnetic molecule $\left\{{\mathrm{V}}_6\right\}$ in [Phys. Rev. Lett. 94, 147204 (2005)]. An analogous equation is derived for the magnetization, at low temperatures, of antiferromagnetic ring systems. The LZS effect is discussed for magnetic molecules with a low spin ground state, for which we arrive at a very convenient set of equations that take into account the combined effects of LZS and thermal transitions. In particular, these equations explain the deviation from exact magnetization reversal at $B\approx 0$ observed in $\left\{{\mathrm{V}}_6\right\}$. They also account for the small magnetization plateaus (“magnetic Foehn effect”), following the LZS steps that have been observed in several magnetic molecules. Finally, we discuss the role of the phonon bottleneck effect at low temperatures and specifically we indicate how this can give rise to a pronounced Foehn effect.

Figure 1

(Color online) The hysteretic behavior of AFM ring systems at low $T$ and for two commonly used sweep forms $B\left(t\right)$ shown in Fig. 1b. The loops are obtained by a numerical solution of Eq. (25), with $M_{\mathit{eq}}$ given in the text and $1∕{\tau }_s$ taken as the sum of the one-phonon term plus a small residual contribution. All the parameters used are arbritrary. The equilibrium magnetization is denoted by the dashed line. The low-energy diagram and the true level crossing between the two relevant states ∣0,0⟩ and $\mid 1,-1⟩$ is shown in Fig. 1c.

Figure 2

(Color online) The hysteresis and LZS effect at $B\approx 0$ for the case of an $S=1∕2$ magnetic molecule, as obtained by solving numerically our equations (see text) for the full-cycle sweep shown in Fig. 2b. The shaded area indicates the LZS regime. The equilibrium magnetization is denoted by the dashed line. Note the deviation from the exact magnetization reversal at $B\approx 0$ as described in the text, and the formation of the small plateau after the step. $B^{*}$ denotes the field at which $M$ crosses $M_{\mathit{eq}}$, as discussed in the text. The low-energy diagram and the level anticrossing between the two levels $\mid -1∕2⟩$ and $\mid +1∕2⟩$ is shown in Fig. 2c. The small circles in Figs. 2b, 2c indicate the LZS regimes, where the magnetization steps take place.

Figure 3

(Color online) The hysteresis and LZS effect at $B\approx B_c$ for AFM ring systems at low $T$ and for fields around the first level-crossing value $B_c$, as obtained by solving numerically our equations (see text) for the half-cycle sweep shown in Fig. 3b. The shaded area indicates the LZS regime. The equilibrium magnetization is denoted by the dotted line. Note the deviation from the pure quantum mechanical prediction regarding the step (see text) and the formation of small plateaus after each step. $B^{*}$ denotes the fields at which $M$ crosses $M_{\mathit{eq}}$, as discussed in the text. The low-energy diagram and the level anticrossing between the levels ∣0,0⟩ and $\mid 1,-1⟩$ is shown in Fig. 3c. The small circles in Figs. 3b, 3c indicate the LZS regimes, where the magnetization steps take place. This typical behavior of $M\left(t\right)$ in the case of $\delta \ne 0$ should be contrasted with that of $\delta =0$ shown in Fig. 1.

Figure 4

(Color online) The magnetic Foehn effect for a magnetic molecule with a spin $S=1∕2$ ground state, for fixed $T$ and for three different fields sweeps with constant sweep rates $r$ (in arbritrary units). The shaded area denotes the LZS regime. Because of the LZS effect, immediately after exiting the LZS regime (shaded area) $M_{\mathit{out}}>M_{\mathit{eq}}$, as opposed to a usual hysteresis situation. Then, outside the LZS regime, where $M$ tends towards $M_{\mathit{eq}}$ (see text), $M$ drops even though the field increases; a local minimum of $M$ is formed at a field $B^{*}$ at which $M=M_{\mathit{eq}}$. Note that this minimum is departing from the LZS regime of the magnetization step and broadens (eventually giving rise to a plateau) with increasing sweep rates.