Landau-Zener tunneling in the presence of weak intermolecular interactions in a crystal of ${\mathrm{Mn}}_4$ single-molecule magnets

A ${\mathrm{Mn}}_4$ single-molecule magnet (SMM), with a well-isolated spin ground state of $S=9∕2$, is used as a model system to study Landau-Zener (LZ) tunneling in the presence of weak intermolecular dipolar and exchange interactions. The anisotropy constants $D$ and $B$ are measured with minor hysteresis loops. A transverse field is used to tune the tunnel splitting over a large range. Using the LZ and inverse LZ method, it is shown that these interactions play an important role in the tunnel rates. Three regions are identified: (i) at small transverse fields, tunneling is dominated by single tunnel transitions, (ii) at intermediate transverse fields, the measured tunnel rates are governed by reshuffling of internal fields, and (iii) at larger transverse fields, the magnetization reversal starts to be influenced by the direct relaxation process, and many-body tunnel events may occur. The hole digging method is used to study the next-nearest-neighbor interactions. At small external fields, it is shown that magnetic ordering occurs which does not quench tunneling. An applied transverse field can increase the ordering rate. Spin-spin cross-relaxations, mediated by dipolar and weak exchange interactions, are proposed to explain additional quantum steps.

Figure 1

(Color online) (a) Unit cells of the ${\mathrm{Mn}}_4$ crystal. Only the cores of the ${\mathrm{Mn}}_4$ molecules are shown. The largest spheres are Mn, the smallest Si, and the others O atoms. The distances between next-nearest molecules are indicated. (b) Schematic of the chainlike coupling between the Mn4 SMMs. (c) Scheme of the chain model where the $S=9∕2$ spin of each molecule is represented by an arrow.

Figure 2

(Color online) Zeeman diagram of the 10 levels of the $S=9∕2$ manifold of ${\mathrm{Mn}}_4$ as a function of the field applied along the easy axis. The spin Hamiltonian parameters are $D=0.608\mathrm{K}$, $B=3.8\mathrm{mK}$, and $E=0.032\mathrm{K}$. The levels are approximately labeled with quantum numbers $m=\pm 9∕2,\pm 7∕2,\dots ,\pm 1∕2$.

Figure 3

(Color online) Detail at a level crossing $m$ with $m^{\prime }$ where the transverse terms (terms containing $S_x$ or/and $S_y$ spin operators) turn the crossing into an avoided level crossing. The initial state is the lower energy state in (a) (standard LZ method) whereas in (b) it is the higher energy state (inverse LZ method).

Figure 4

(Color online) (a) Hysteresis loop measurements of a single crystal of ${\mathrm{Mn}}_4$ at low temperatures $\left(40\mathrm{mK}\right)$ where thermal activation to excited spin states can be neglected. The field is applied in the direction of the easy axis of magnetization and swept at a constant rate between 0.002 and $0.14\mathrm{T}∕\mathrm{s}$. The dots labeled with $M_{\mathrm{FC}}$ are the magnetization after cooling the sample from $5\mathrm{K}$ down to $0.04\mathrm{K}$ in a constant applied field $H_z$. $M_{\mathrm{FC}}$ is used for the equilibrium magnetization $M_{\mathrm{eq}}$ in Sec. 4A. (b) Hysteresis loop measurements similar to Fig. 4a but at different temperatures and for a field sweep rate of $0.14\mathrm{T}∕\mathrm{s}$.

Figure 5

(Color online) Minor hysteresis loops of a single crystal of ${\mathrm{Mn}}_4$. The magnetization was first saturated at $-1.4\mathrm{T}$. After ramping the field to zero at $0.14\mathrm{T}∕\mathrm{s}$, the field was swept $k$ times back and forth [between 0.028 and $0.07\mathrm{T}$ in (a) and between $-0.028$ and $0.07\mathrm{T}$ in (b)] over the zero-field resonance with a sweep rate of $0.014\mathrm{T}∕\mathrm{s}$. After the $k\text{th}$ sweep, the field is quickly swept back to $-1.4\mathrm{T}$ at a rate of $0.14\mathrm{T}∕\mathrm{s}$ leading to resonant tunneling at the level crossing $(m,m^{\prime })=(-7∕2,9∕2)$ and $(-5∕2,9∕2)$, and allow to determine ${\mu }_0{H}_z^{\left(1\right)}=0.544\mathrm{T}$ and ${\mu }_0{H}_z^{\left(2\right)}=1.054\mathrm{T}$. The field interval of the $k$ back and forth sweeps corresponds to zero reversed neighbor (0 RN, see Sec. 4B1) in (a) whereas it goes over the 0 RN, 1 RN, and 2 RN transitions in (b). Note that the procedure in (a) leads to sharper steps and reduce spin-spin cross-relaxtion (SSCR) (Sec. 5) because all reversed spins have two nonreversed neighbors. The transitions of SSCR are indicated in (b).

Figure 6

Calculated tunnel splitting for ${\mathrm{Mn}}_4$ using $D=0.608\mathrm{K}$, $B=3.8\mathrm{mK}$, and $E=0.038\mathrm{K}$ (Ref. 42). The calculated $\Delta$ has been averaged over all possible orientations of the transverse field in order to represent the arbitrary orientation of the $E$ term. The influence of nuclear spin broadening was taken into account by a Gaussian distribution of transverse field components with a half-width $\sigma =0.035\mathrm{T}$.

Figure 7

(Color online) Calculated LZ tunnel probabilities $P_{\pm 9∕2}$ as a function of transverse field using the tunnel splitting from Fig. 6 and the indicated field sweep rates. Only every 100th calculated point is shown as a symbol.

Figure 8

(Color online) Magnetization vs applied field scans for (a)–(d) the LZ method and (e)–(h) the inverse LZ method. The indicated transverse fields $H_{\mathrm{trans}}$ are applied at point 1.

Figure 9

(Color online) The change of magnetization $\Delta M=(M_{\mathrm{f}}-M_{\mathrm{i}})(dH∕dt)∕\mid dH∕dt\mid$ as a function of transverse field $H_{\mathrm{trans}}$ for several LZ field sweeps; (a) LZ method and (b) inverse LZ method. Only every 30th measured point is shown as a symbol.

Figure 10

(Color online) The change of magnetization $\Delta M=(M_{\mathrm{f}}-M_{\mathrm{i}})(dH∕dt)∕\mid dH∕dt\mid$ as a function of transverse field $H_{\mathrm{trans}}$ for the LZ field sweep from $\mathbf{1}$ to $\mathbf{2}$ and for the indicated field sweep rates; (a) LZ method and (b) inverse LZ method.

Figure 11

Scheme of the hole digging method presenting the time dependence of temperature, applied field, and magnetization of the sample.

Figure 12

(Color online) Typical square root of time relaxation curves for an ${\mathrm{Mn}}_4$ crystal measured at $40\mathrm{mK}$. For each curve, the sample was first (a) saturated or (b) thermally annealed at $H=0$. Then the indicated field was applied and the short-time relaxation of magnetization was measured. The slope of the lines gives ${\Gamma }_{\mathrm{sqrt}}$ when plotted against the square root of $t$ as shown.

Figure 13

(Color online) Field dependence of the short time square-root relaxation rates ${\Gamma }_{\mathrm{sqrt}}\left(H_z\right)$ for three different values of the initial magnetization $M_{\mathrm{in}}$. According to Eq. (8), the curves are proportional to the distribution $P\left(H_z\right)$ of magnetic energy bias due to local internal field distributions in the sample. Note the logarithmic scale for ${\Gamma }_{\mathrm{sqrt}}$. The peaked distribution labeled $M_{\mathrm{in}}=-0.99M_{\mathrm{s}}$ was obtained by saturating the sample, whereas the other distributions were obtained by thermal annealing. For $M_{\mathrm{in}}⪡M_{\mathrm{s}}$, the curves are distorted by nearest-neighbor effects. The peak at $\pm 0.036$ and $0\mathrm{T}$ are from molecules which have zero, one, or two nearest-neighbor (RN) molecules with reversed magnetization.

Figure 14

(Color online) The field dependence of the short-time square-root relaxation rates ${\Gamma }_{\mathrm{sqrt}}\left(H_z\right)$ are presented on a logarithmic scale showing the depletion of the molecular spin states by quantum tunneling at $H_{\mathrm{dig}}=0.028\mathrm{T}$ for various digging times $t_{\mathrm{dig}}$ and $M_{\mathrm{in}}=-M_s$.

Figure 15

(Color online) Schematical representation of a chain of spins with dipole and exchange coupling represented by + and − poles and $\pm J$, respectively. (a) All spins have zero reversed neighbors (0 RN); in (b) and (c), the indicated spins have one reversed neighbor (1 RN), whereas in (d) it has two reversed neighbors (2 RN). The three cases of this two-neighbor-model lead to the fine structure of three for all quantum resonance steps.

Figure 16

(Color online) Scheme of the distributions of internal fields of (a) a randomly ordered, (b) a partially ordered, and (c) a completely ordered state with zero total magnetization. Here, $N_{+}$ and $N_{-}$ are the distributions for up and down spins, respectively. The fine structure with three bumps are due to three cases of zero reversed neighbors (0 RN), one reversed neighbor (1 RN), and two reversed neighbors (2 RN). The broadening is due to long-range dipolar fields of an arbitrary (not ellipsoidal) crystal shape.

Figure 17

(Color online) Short-time square-root relaxation rate for an ${\mathrm{Mn}}_4$ crystal measured at $40\mathrm{mK}$. (a) A randomly disordered state with $M_{\mathrm{in}}=0$ was first created by sweeping 100 times the field back and forth over the zero-field resonance with a sweep rate of $0.14\mathrm{T}∕\mathrm{s}$. After a waiting time $t_{\mathrm{w}}$ at $H=0$, the short-time relaxation rate was measured. The field for one reversed neighbor (1 RN) is indicated. (b) First, the sample was cooled at $H=0$. Then, a transverse field of $1.12\mathrm{T}$ was applied. After a waiting time $t_{\mathrm{w}}$ the transverse field was set to zero. After another waiting time $t_{\mathrm{w}}$ at $H=0$, the short-time relaxation rate was measured.

Figure 18

(Color online) Minor hysteresis loops of a single crystal of ${\mathrm{Mn}}_4$. The magnetization was first saturated at $-1.4\mathrm{T}$. After ramping the field to zero, the field was swept several times back and forth over a part of the tunneling step at the level crossing $(m,m^{\prime })=(-9∕2,7∕2)$: for curve (1) between 0.38 and $0.62\mathrm{T}$, for curve (2) between 0.38 and $0.5\mathrm{T}$, and for curve (3) between 0.52 and $0.62\mathrm{T}$. After the back and forth sweeps, the field is swept to $1.4\mathrm{T}$. The field sweep rate for all parts is $0.07\mathrm{T}∕\mathrm{s}$. The fine structure due to nearest neighbor effects and SSCR are indicated.