Berry-phase effects in transport through single Jahn-Teller molecules

The vibrational modes of Jahn-Teller molecules are affected by a Berry phase that is associated with a conical intersection of the adiabatic potentials. We investigate theoretically the effect of this Berry phase on the electronic transport properties of a single $E\otimes e$ Jahn-Teller molecule when the tunneling electrons continually switch the molecule between a symmetric and a Jahn-Teller distorted charge state. We find that the Berry phase, in conjunction with a spectral trapping mechanism, leads to a current-blockade even in regions outside the Coulomb blockade. The blockade is strongly asymmetric in the gate voltage and induces pronounced negative differential conductance.

Figure 1

(Color online) (Top): Adiabatic potentials of the $E\otimes e$ Jahn-Teller distorted (charged) molecule (left) and the undistorted (neutral) molecule (right) as a function of $(Q_{\theta },Q\epsilon )$. (Bottom): Displacement patterns of the degenerate $\theta$ and $\epsilon$ vibrations for octahedral ${\mathrm{ML}}_6$ molecules.

Figure 2

(Color online) Vibrational spectra (a) of the neutral and (b) of the charged molecule vs pseudoangular momentum. Only energies for positive pseudoangular momenta are shown as the spectra are symmetric under the reflection $l\mapsto -l$.

Figure 3

Differential conductance vs gate voltage $V_{\mathrm{g}}$ and source-drain voltage $V_{\mathrm{sd}}$ for $\lambda =1$, $k_{\mathrm{B}}T=0.01\hslash \omega$. (a) $\tau =10{\tau }_0$; (b) $\tau =5000{\tau }_0$. White corresponds to positive and black to negative values. Neutral gray is zero differential conductance.

Figure 4

(Color online) Angular-momentum resolved spectra of the Jahn-Teller molecule $(\lambda =1)$ (diamonds) and the neutral molecule (circles). Radial excitations are grouped for each pseudoangular momentum. Panel (a) shows the situation for tunneling at the source electrode and panel (b) the situation at the drain electrode. The chosen parameters are ${\mathit{eV}}_{\mathrm{sd}}=2\hslash \omega$ and ${\mathit{eV}}_{\mathrm{g}}=-0.5\hslash \omega$. Consider panel (a). An electron can only tunnel from the source electrode and induce the vibronic transition $\mid 0;q⟩\to \mid 1;q^{\prime }⟩$ if the difference in the electronic energy, $\frac{1}{2}{\mathit{eV}}_{\mathrm{sd}}-{\mathit{eV}}_{\mathrm{g}}$, is larger than the difference in the vibronic energy, $E_{1q^{\prime }}-E_{0q}$. More precisely, the Fermi function $f(e(V_{\mathrm{g}}-\frac{1}{2}{V}_{\mathrm{sd}})+E_{1q^{\prime }}-E_{0q})$ has to be positive. By shifting the neutral spectrum by $\frac{1}{2}e{\mathit{V}}_{\mathrm{sd}}$ and the charged spectrum by ${\mathit{eV}}_{\mathrm{g}}$, this relation translates into the intuitive rule that in the diagram the target state of a tunnel process has to lie below the initial state. Energetically allowed transitions are therefore directed downward in the figure. A horizontal transition corresponds to the borderline case of an electron sitting exactly at the Fermi level of the electrode. That conservation of energy has the form of an inequality is due to the filled continuum of states in the Fermi-gas electrode. Panel (b) shows the corresponding energetic situation for sequential tunneling processes at the drain electrode. The presence of trapping states can be seen clearly.

Figure 5

(Color online) Angular-momentum resolved spectra for tunneling to the drain electrode, ${\mathit{eV}}_{\mathrm{sd}}=2\hslash \omega$ and ${\mathit{eV}}_{\mathrm{g}}=-0.5\hslash \omega$. The trapping state $\mid \beta ⟩=\mid 1;\frac{5}{2},0⟩$ is marked in red. The three nearest energy manifolds of the uncharged system are shaded in gray. The dashed arrows show transitions that are forbidden, because the target manifold’s energy is too large. The dotted arrow shows a transition that is forbidden due to the selection rule.

Figure 6

(Color online) Angular-momentum resolved spectra like in Fig. 4. In this case the Jahn-Teller molecule is placed slightly below the chemical potential of the source electrode. The chosen parameters are ${\mathit{eV}}_{\mathrm{sd}}=2\hslash \omega$ and ${\mathit{eV}}_{\mathrm{g}}=0.5\hslash \omega$. Contrary to the situation shown in Fig. 4, there are no trapping states.

Figure 7

(Color online) Positive differential conduction peak at the Coulomb blockade, $V_{\mathrm{g}}=-0.5\hslash \omega$, for different values of the relaxation time $\tau$ (a) and fit with the simplified model, dots in (b). The red curves, that is, the large, centered peaks have small, and the blue ones, damped and shifted to lower $V_{\mathrm{sd}}$, very large $\tau$.

Figure 8

(Color online) Fit $\tau$ values of the simplified model (dots) and its application to Fig. 7 vs the numerical values used in the actual simulation. The curve is a linear fit to these values, ${\tau }_{\text{fit}}\propto {\tau }_{\text{numerical}}$, showing the validity of the simplified approach in the regime of strong separation of the time scales.