Tight-binding model of Mn${}_{12}$ single-molecule magnets: Electronic and magnetic structure and transport properties

We describe and analyze a tight-binding model of single-molecule magnets (SMMs) that captures both the spin and spatial aspects of the SMM electronic structure. The model generalizes extended Hückel theory to include the effects of spin polarization and spin-orbit coupling. For neutral and negatively charged Mn${}_{12}$ SMMs with acetate or benzoate ligands, the model yields the total SMM spin, the spins of the individual Mn ions, the magnetic easy axis orientation, the size of the magnetic anisotropy barrier, and the size of the highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO-LUMO) gap consistent with experiment. For neutral molecules, the predicted spins and spatial locations of the HOMO are consistent with the results of density functional calculations. For the total spin and location of the LUMO, density functional theory based calculations yield varied results, while the present model yields results consistent with experiments on negatively charged molecules. For Mn${}_{12}$ SMMs with thiolate- and methylsulphide-terminated benzoate ligands (Mn${}_{12}$-Ph-Th), we find the HOMO to be located on the magnetic core of the molecule, but (unlike for the Mn${}_{12}$ SMMs that have previously been studied theoretically) we predict the LUMO and near-LUMO orbitals of Mn${}_{12}$-Ph-Th to be located on ligands. Therefore, we predict that for these Mn${}_{12}$ SMMs, resonant and off-resonant coherent transport via near-LUMO orbitals, not subject to Coulomb blockade, should occur. We propose that this effect can be used to identify specific experimentally realized SMM transistors in which the easy axis and magnetic moment are approximately parallel to the direction of the current flow. We also predict effective spin filtering by these SMMs to occur at low bias whether the transport is mediated by the HOMO that is on the magnetic core of the SMM or by near-LUMO orbitals located on the nominally nonmagnetic ligands.

Figure 1

Calculated energy of the neutral Mn${}_{12}$-Ac molecule with ground-state spin $S_{\text{total}}=10$ versus the total spin projected on the easy axis. The arrow shows the magnetic anisotropy barrier.

Figure 2

Projected density of states for majority spin (up) and minority spin (down) electrons on the oxygen, inner Mn, outer Mn, and carbon atoms for Mn${}_{12}$-Ac.

Figure 3

Projected density of states for majority spin (up) and minority spin (down) electrons on the oxygen, inner Mn, outer Mn, and carbon atoms for Mn${}_{12}$-Ph.

Figure 4

Calculated energies of Mn${}_{12}$-Ph versus total spin projected onto the easy axis for (a) neutral molecule with spin ground state $S_{\text{total}}=10$, (b) unrelaxed (explained in text) negatively charged molecule with $S_{\text{total}}=9\frac{1}{2}$, and (c) realistic (explained in text) negatively charged molecule with $S_{\text{total}}=9\frac{1}{2}$. Arrows show the magnetic anisotropy barriers.

Figure 5

Two views of the Mn${}_{12}$-Ph-Th SMM. The ligands are terminated with methylthio (SCH${}_3$) groups. The Mn atoms are located near the $x$-$y$ plane and the magnetic easy axis is aligned with the $z$ axis. We refer to the ligands that are close to the $x$-$y$ plane as “in-plane ligands” and to the others as “out-of-plane ligands.” The molecule is assumed to be located between two gold leads (not shown) and to bond to each of the gold leads via the sulfur atom of a single ligand, the methyl group having been removed from that sulfur atom. The pairs of open circles with the same color indicate pairs of sulfur atoms that are attached to the gold leads in the different bonding configurations that we consider. (a) The red, blue, green, orange, purple, and brown circles labeled IPB${}_i$ indicate the pair of sulfur atoms bonding to the gold in the $i\mathrm{th}$ bonding configuration involving in-plane ligands. (b) The red, green, purple, and orange circles labeled OPB${}_i$ indicate the pair of sulfur atoms bonding to the gold in the $i\mathrm{th}$ bonding configuration involving out-of-plane ligands. Atoms are color labeled: manganese (red), carbon (gray), sulfur (yellow), oxygen (blue), and hydrogen (white).

Figure 6

Projected density of states for majority spin (up) and minority spin (down) on the sulfur, oxygen, inner Mn, outer Mn, and carbon atoms for Mn${}_{12}$-Ph-Th.

Figure 7

Total transmission probability at zero bias and gate voltage as a function of energy $E$ relative to Fermi energy $E_F$ of gold for the different in-plane bonding geometries between the molecule and leads that are identified in Fig. 5a. The inset is a schematic of the SMM and electrodes. The magnetic core of the molecule that contains the Mn atoms is pink, the arrow indicates the magnetic easy axis, and the electrodes are yellow.

Figure 8

Total transmission probability at zero bias and gate voltage as a function of energy $E$ relative to Fermi energy $E_F$ of gold for the different out-of-plane bonding geometries between the molecule and leads that are identified in Fig. 5b. The inset is a schematic of the SMM and electrodes. The magnetic core of the molecule that contains the Mn atoms is pink, the arrow indicates the magnetic easy axis, and the electrodes are yellow.

Figure 9

Calculated current [from Eq. (9)] as a function of bias voltage at zero gate voltage and temperature for different bonding configurations of the SMM relative to electrodes identified in Fig. 5.

Figure 10

Projected density of states (PDOS) on carbon atoms for energies close to the LUMO and related wave-function isosurfaces for Mn${}_{12}$-Ph-Th. The energies of the states with wave-function isosurfaces (a), (b), and (c) are indicated by arrows in the PDOS plot. The LUMO + 5, LUMO + 14, and LUMO + 17 molecular orbitals are marked in the PDOS plot with filled red circles.

Figure 11

Projected density of states (PDOS) on outer Mn atoms in an energy range close to the HOMO and related wave-function isosurfaces for Mn${}_{12}$-Ph-Th. The energies of the states with wave-function isosurfaces (a), (b), and (c) are indicated by arrows in the PDOS plot.

Figure 12

(a)–(d) Calculated spin-resolved transmission probabilities at zero bias and gate voltage as a function of energy $E$ relative to Fermi energy $E_{\text{F}}$ for a Mn${}_{12}$-Ph-Th molecule bridging a pair of gold electrodes via in-plane ligands in the bonding configuration IPB${}_1$ defined in Fig. 5a. Transmission of (a) spin up to spin up, (b) spin down to spin down, (c) spin up to spin down, (d) spin down to spin up. Note that the peaks visible in (c) and (d) are a direct consequence of spin-orbit coupling. The insets show the spin-up (-down) to spin-up (-down) transmission probability at zero bias and gate voltage magnified by a factor of 500 relative to the plots in panels (a) and (b); the tails of the spin-down to spin-down transmission peaks are responsible for the smooth onset of the spin-down current with increasing bias in (e). The calculated spin-resolved current is also shown for (e) positive and (f) negative gate voltage as a function of bias voltage at zero temperature. Solid red (dashed blue) lines represent spin-up (-down) current. (e) At low bias voltages for positive gate voltages (gate potential at the molecule $=+0.2$ V), only spin-down electrons are transmitted for bias voltages less than 0.65 eV. (f) For the negative gate voltage (gate potential at the molecule $=-0.2$ V), only spin-up electrons are transmitted.

Figure 13

(a)–(d) Calculated spin-resolved transmission probabilities at zero bias and gate voltage as a function of energy $E$ relative to Fermi energy $E_{\text{F}}$ for a Mn${}_{12}$-Ph-Th molecule bridging a pair of gold electrodes via out-of-plane ligands in the bonding configuration OPB${}_1$ defined in Fig. 5a. Transmission of (a) spin up to spin up, (b) spin down to spin down, (c) spin up to spin down, (d) spin down to spin up. Note that the peaks visible in (c) and (d) are a direct consequence of spin-orbit coupling. Calculated spin-resolved current for (e) positive and (f) negative gate voltage as a function of bias voltage at zero temperature. Solid red (dashed blue) lines represent spin-up (-down) current. (e) For positive gate voltages (gate potential at the molecule $=+0.2$ V) at low bias voltage (0 $<$V${}_{\text{bias}}<$0.53 V), most of the transmitted electrons are spin up. For $V_{\text{bias}}>$0.53 V, the current is not spin polarized. (f) For negative gate voltages (gate potential at the molecule $=-0.2$ V) at low bias voltage, only spin-up electrons are transmitted.