Cotunneling spectroscopy and the properties of excited-state spin manifolds of ${\mathrm{Mn}}_{12}$ single molecule magnets

We study charge transport through single molecule magnet (SMM) junctions in the cotunneling regime as a tool for investigating the properties of the excited-state manifolds of neutral ${\mathrm{Mn}}_{12}$ SMs. This study is motivated by a recent transport experiment [S. Kahle et al., Nano Lett. 12, 518 (2012)] that probed the details of the magnetic and electronic structure of ${\mathrm{Mn}}_{12}$ SMMs beyond the ground-state spin manifold. A giant spin Hamiltonian and master equation approach is used to explore theoretically the cotunneling transport through ${\mathrm{Mn}}_{12}$-Ac SMM junctions. We identify SMM transitions that can account for both the strong and weak features of the experimental differential conductance spectra. We find the experimental results to imply that the excited spin-state manifolds of the neutral SMM have either different anisotropy constants or different $g$ factors in comparison with its ground-state manifold. However, the latter scenario accounts best for the experimental data.

Figure 1

Second derivative of the current, $\frac{d^{2}I}{d{V}^2}$, through the ${\mathrm{Mn}}_{12}$ SMM as a function of the bias voltage calculated based on the GSH. All of the spin manifolds have the same anisotropy constants $D$ and magnetic $g$ factors. The red (green) curve represents $\frac{d^{2}I}{d{V}^2}$ at the applied magnetic field $B=0\text{T}$ $(B=10\text{T}$ parallel to the SMM easy axis). The parameters of the GSH are $S_{\text{tot}}^{\text{g}}=10,S_{\text{tot}}^{\text{e}}=9,D^{\text{g}}/{\hslash }^2=D^{\text{e}}/{\hslash }^2=0.06$ meV, ${\epsilon }_0=1$ eV, $T=1.5$ K, $g^{\text{g}}=g^{\text{e}}=2$, and ${\gamma }_R/{\gamma }_L={10}^{-4}$. (b), (c) Schematic diagram representing the energies of the two manifolds of the neutral SMM $(S_{\text{tot}}=10$ and $S_{\text{tot}}=9)$ and negatively charged SMM. (b) The blue arrows show the inelastic cotunneling processes within the ground-Ðstate manifold. The electron tunnels into the SMM which is in its ground state of the ground-state manifold, $|10,10\rangle$, and tunnels out of the SMM leaving the SMM in its first excited state of the ground-state manifold, $|10,9\rangle$. This transport process is mediated by the ground state of the ground-state manifold $|9\frac{1}{2},9\frac{1}{2}\rangle$ of the negatively charged SMM that is the virtual state. Transitions mediated by states in the $S=9$ manifold do not occur because the $S=9$ manifold has the same charge as the $S=10$ manifold and therefore cannot be an intermediate (virtual) state in the cotunneling process. (c) The red arrows show the inelastic cotunneling processes between the two manifolds of the neutral SMM. The electron tunnels into the SMM which is initially in its ground state of the ground-state manifold, $|10,10\rangle$, and tunnels out of the SMM leaving the SMM in its ground state of the first-excited manifold, $|9,9\rangle$. This process is mediated by the ground state of the ground–state manifold $|9\frac{1}{2},9\frac{1}{2}\rangle$ of the negatively charged SMM as the virtual state. The direct transition from the $|10,10\rangle$ state to the $|9,9\rangle$ state is expected to have a lower probability since it would involve direct tunneling of an electron from one electrode to the other, effectively via the vacuum level instead of making use of virtual molecular states that are at lower energies.

Figure 2

The energy difference (a) ${\Delta }_{|10,10\rangle ,|10,9\rangle }$ and [(b), (c)] ${\Delta }_{|10,10\rangle ,|9,9\rangle }$ as a function of the applied magnetic field calculated by numerical diagonalization of the GSH for different values of $\theta$, the angle between the applied magnetic field and the SMM easy axis, assuming that $\theta$ and $g$ are the same for all of the manifolds. For simplicity, the energy differences ${\Delta }_{|10,10\rangle ,|10,9\rangle }$ and ${\Delta }_{|10,10\rangle ,|9,9\rangle }$ are set to zero at $B=0$ T. The parameters of the GSH are (a)–(c) $S_{\text{tot}}^{\text{g}}=10,S_{\text{tot}}^{\text{e}}=9,{\epsilon }_0=1$ eV, $g^{\text{g}}=g^{\text{e}}=2$; (b) $D^{\text{g}}/{\hslash }^2=D^{\text{e}}/{\hslash }^2=0.06$ meV; and (c) $D^{\text{e}}/{\hslash }^2=0.02$ meV.

Figure 3

(a) Second derivative of the current, $\frac{d^{2}I}{d{V}^2}$, through the ${\mathrm{Mn}}_{12}$ SMM as a function of the bias voltage. The red (blue) curve represents $\frac{d^{2}I}{d{V}^2}$ at the applied magnetic field $B=0$ T $(B=10$ T with $\theta ={60}^{\circ })$. In this calculation the neutral SMM has two manifolds. The parameters of the GSH are as in Fig. 2. (b) Schematic diagram representing the energies of the two manifolds of the neutral SMM $(S_{\text{tot}}=10$ and $S_{\text{tot}}=9)$ under an applied magnetic field. The green arrows show the allowed transitions from the ground state of the ground-state manifold when the applied magnetic field is parallel to the SMM easy axis. The blue arrows show the additional allowed transitions from the ground state of the ground-state manifold when the applied magnetic field is not parallel to the SMM easy axis.

Figure 4

Second derivative of the current, $\frac{d^{2}I}{d{V}^2}$, of the ${\mathrm{Mn}}_{12}$ SMM as a function of the bias voltage calculated based on the GSH assuming different values of the g factor for the ground- and excited-state manifolds. The red curve represents $\frac{d^{2}I}{d{V}^2}$ at the applied magnetic field $B=0$ T. The green (blue) curve represents $\frac{d^{2}I}{d{V}^2}$ at the applied magnetic field $B=10$ T parallel to the SMM easy axis $(B=10$ T with a $\theta ={75}^{\circ })$. $\theta$ is the tilting angle between the applied magnetic field and the SMM easy axis. The parameters of the GSH are as in Fig. 1 except that $g^{\text{g}}/g^{\text{e}}=1.2$.

Figure 5

Calculated second derivative of the current, $\frac{d^{2}I}{d{V}^2}$, through the ${\mathrm{Mn}}_{12}$ SMM as a function of the bias voltage for $B=0$ and 10 T. The magnetic field is parallel to the magnetic easy axis and the parameters of the GSH are as in Fig. 4 except that ${\gamma }_R/{\gamma }_L={10}^{-2}$.