Kondo effect in single-molecule magnet transistors

We present a careful and thorough microscopic derivation of the anisotropic Kondo Hamiltonian for single-molecule magnet (SMM) transistors. When the molecule is strongly coupled to metallic leads, we show that by applying a transverse magnetic field it is possible to topologically induce or quench the Kondo effect in the conductance of a SMM with either an integer or a half-integer spin $S>1/2$. This topological Kondo effect is due to the Berry-phase interference between multiple quantum tunneling paths of the spin. We calculate the renormalized Berry-phase oscillations of the two Kondo peaks as a function of the transverse magnetic field by means of the poor man’s scaling. In particular, we show that the Kondo exchange interaction between itinerant electrons in the leads and the SMM pseudospin 1/2 depends crucially on the SMM spin selection rules for the addition and subtraction of an electron and can range from antiferromagnetic to ferromagnetic. We illustrate our findings with the SMM ${\text{Ni}}_4$, which we propose as a possible candidate for the experimental observation of the conductance oscillations.

Figure 1

(Color online) Schematic illustration of a single-molecule field-effect transistor formed by a SMM attached to two metallic leads (source and drain) and controlled by a back gate voltage.

Figure 2

(Color online) Energy spectra of consecutive charging sectors of the SMM Hamiltonian $\left(q=-1,0,1\right)$. ${\Delta }_q\equiv {\Delta }_{m_q,-m_q}$ is the tunnel splitting due to the in-plane anisotropy and $E_{s/a,m_q}^{\left(q\right)}$ denote energy eigenstates corresponding to symmetric and antisymmetric combinations of the ground eigenstates $|\pm s_q⟩$ of the longitudinal component of the SMM total spin $S_{q,z}$.

Figure 3

(Color online) Diagrams representing the (a) longitudinal and (b) transversal exchange interactions between the SMM magnetization and the electrons in the leads.

Figure 4

The renormalization of the $g$ factor due to the Knight shift. As an estimate, we use ${\rho }_0{J}_{\pm }={\rho }_0{J}_z=0.15$, where ${\rho }_0=9.45\times {10}^{20}{\text{J}}^{-1}$(see Ref. 26).

Figure 5

The graph shows the temperature dependence of the zeros for the spin ground state of the SMM ${\text{Ni}}_4$ due to the Berry-phase oscillation as a function of the transverse magnetic field for the tunnel splittings between $|4⟩$ and $|-4⟩$ states $\left(S_0=4\right)$ and the following temperature values: (a) $T/T_K=1.5$, (b) $T/T_K=1.6$, (c) $T/T_K=1.7$, and (d) $T/T_K=1.8$. We used the following values for the anisotropy constants $A_q=0.00011\text{eV}$, $B_{2,q}=2.96\mu \text{eV}$, and $B_{4,q}=-0.25\mu \text{eV}$.

Figure 6

Diagrams showing the asymmetric cut of the left- and right-contact bands when a finite bias is applied.

Figure 7

Plots showing the temperature dependence of the conductance and the zeros for the tunnel splittings between the spin ground states $|4⟩$ and $|-4⟩$ of the SMM ${\text{Ni}}_4$ $\left(S_0=4\right)$ due to the Berry-phase oscillation as a function of the transverse magnetic field for the following temperature values: (a) $T/T_K=1.5$, (b) $T/T_K=1.6$, (c) $T/T_K=1.7$, and (d) $T/T_K=1.8$. We used a bias voltage of $V_b=V_L-V_R=2.7\mu \text{V}$ and the following values for the anisotropy constants $A_q=0.00011\text{eV}$, $B_{2,q}=2.96\mu \text{eV}$, and $B_{4,q}=-0.25\mu \text{eV}$.

Figure 8

(Color online) Graph showing ${\text{log}}_{10}I$ versus the transverse magnetic field for $V_b=V_L-V_R=4\times {10}^{-3}\text{eV}$, $V_g=0.01\text{eV}$, and ${\gamma }_{4,-4}={10}^{12}{\text{s}}^{-1}$. One can see that the current is suppressed at the zeros of the tunnel splittings ${\Delta }_1$ and ${\Delta }_0$.