Dynamical magnetic anisotropy and quantum phase transitions in a vibrating spin-1 molecular junction

We study the electronic transport through a spin-1 molecule in which mechanical stretching produces a magnetic anisotropy. In this type of device, a vibron mode along the stretching axis will couple naturally to the molecular spin. We consider a single molecular vibrational mode and find that the electron-vibron interaction induces an effective correction to the magnetic anisotropy that shifts the ground state of the device toward a non-Fermi-liquid phase. A transition into a Fermi-liquid phase could then be achieved, by means of mechanical stretching, passing through an underscreened spin-1 Kondo regime. We present numerical renormalization-group results for the differential conductance, the spectral density, and the magnetic susceptibility across the transition.

Figure 1

Phase diagram of a deformable spin-1 molecular break junction in the presence of a local vibrational mode. A rich variety of phases can be found at temperatures below $T_K$, at which Kondo physics dominates the transport properties of the system. The effective anisotropy $A_{\text{eff}}$, induced by both static and dynamic deformations of the molecule, drives the system through different quantum phases.

Figure 2

Spin-1, deformable molecular device. (a) A break junction supporting a spin-1 molecule that can be stretched by mechanical means. (b) The molecule modeled by two coupled orbitals, only one of them connected to metallic leads. The spin-1 regime is enforced by an interorbital ferromagnetic coupling. Magnetic anisotropy, induced by static and dynamic (vibrational) stretching of the molecule, is accounted for by the term $A\left(z\right)S_z^2$ in the Hamiltonian. (c) Lowest-lying level energies of the isolated molecule, obtained for different values of the effective static anisotropy.

Figure 3

(a) Spectral density at orbital $a$ for $A_0=A_1=0$ (reference system). The screening of the molecular triplet ground state by the band is characterized by the sharp Kondo resonance of width $T_K^0$ at the Fermi level ($\omega =0$), followed by shoulders at approximately the singlet-triplet (red dashed lines) and the charge (blue dashed lines in the inset) excitation energies of the isolated molecule. The Kondo effect yields unitary conductance at low ($\ll T_K^0$) temperatures. (b) Same as (a), but for finite magnetic anisotropy ($A_0={\omega }_0=5T_K^0$, $A_1$ from $2T_K^0$ to $11T_K^0$). In the easy-axis regime of $A_{\text{eff}}<0$ (red curves), the system presents a finite amplitude of the spectral density at the Fermi level. The hard-axis regime ($A_{\text{eff}}>0$, blue curves) presents a dip at the Fermi level, keeping the system in a Coulomb blockade. The underscreened Kondo state is recovered when $A_{\text{eff}}=0$ (black curve).

Figure 4

Conductance as a function of temperature, for different values of $A_{\text{eff}}$, (a) varying $A_1$ from $2T_K^0$ to $12T_K^0$ for ${\omega }_0=A_0=5T_K^0$, (b) varying $A_0$ from $4T_K^0$ to $9T_K^0$ for $A_1={\omega }_0=5T_K^0$, and (c) as a function of $A_0$ for different isotherms. The zero-temperature conductance vanishes for positive anisotropy, due to the lack of available states for transport in the molecule at the Fermi level (see Fig. 3). As $A_{\text{eff}}\to 0$ the conductance curves vary as indicated by the blue arrows until the unitary limit is reached, signaled in the plot by a plateau that extends to lower temperatures. Unitarity is then lost due to many-body scattering of the leads' electrons off the $S_z=\pm 1$ degenerate ground state of the molecule at negative anisotropies. As $A_{\text{eff}}$ goes from zero to negative the conductance curves follow the trend indicated by the red arrow. In (c) we show isotherms for the conductance as a function of $A_0$ [corresponding to the curves in (b)], which can be directly related to experiments in which the conductance is measured in a stretchable device, at constant temperature. The quantum critical point becomes apparent at very low temperatures, where a sudden drop in conductance signals the change of sign of $A_{\text{eff}}$. The Kondo temperatures of these cases are lower than the reference scale $T_K^0$ by one order of magnitude, as can be seen by the onset of the conductance plateaus below $T=T_K^0$.

Figure 5

Effective magnetic moment squared, ${\mu }^2\equiv T\chi \left(T\right)$, as a function of temperature for different values of the magnetic anisotropy. Solid lines are for a system with e-ph interaction ($A_1\ne 0$), whereas dashed lines correspond to a system without e-ph interaction, and with a static magnetic anisotropy tuned to match the effective anisotropy of the e-ph interacting system. As expected from the e-ph mediated reduction of the Kondo exchange couplings, the first-stage Kondo temperature is lower in the e-ph interacting system [solid arrows in (b)], that is, $T_K<T_K^0$. This in turn increases the second-stage Kondo temperature $T_K^{*}$ [arrows in (a)], as expected from Eq. (20).

Figure 6

(a) and (c) Magnetic moment squared of the molecule at zero temperature [${\mu }^2(T\to 0)$] as a function of $A_1$ and ${\omega }_0$, for (a) $A_0=5T_K^0$ and (c) $A_0=6T_K^0$. In each case, the red curve ($A_1=\sqrt{{\omega }_0{A}_0}$) indicates the parameters of full degeneracy of the spin-1 triplet in the molecular limit. The black dots indicate the parameters where the underscreened spin-1 Kondo effect is recovered (${\mu }^2=0.25$). (b) and (d) Correction to the magnetic anisotropy due to the coupling to the leads for the parameters of (a) and (c), respectively.

Figure 7

Second-stage Kondo temperature, $T_K^{*}$, for different values of $A_1$ and ${\omega }_0$. (a) An observed fast fall of $T_K^{*}$ as a function of $A_1/{\omega }_0$, consistent with a vibronic induced shift of the magnetic anisotropy and Eq. (20). (b) Plotting $T_K^{*}$ against $A_1^2/{\omega }_0=A_0-\tilde{A}$, making the presence of a correction $\Delta$ to the anisotropy evident; the dotted lines indicate the value of $A_1$ where the molecular limit anisotropy $\tilde{A}$ changes sign, for the corresponding values of $A_0$. The numerical results are clearly shifted toward positive values of $A_{\text{eff}}$. (c) Verification of Eq. (20) within our picture of effective anisotropy and suppressed Kondo exchange couplings, for fixed $A_1$ and ${\omega }_0$, varying $A_0$. We substitute $A_{\text{eff}}$ in the equation, with $\Delta$ computed from numerical results such as those of Figs. 6b and 6d. The slope of each curve is proportional to $-2\sqrt{T_K/T_K^0}$, with $T_K$ the Kondo temperature corresponding to the curve.