Subgap tunneling through channels of polarons and bipolarons in chain conductors

We suggest a theory of internal coherent tunneling in the pseudogap region where the applied voltage is below the free electron gap. We consider quasi-one-dimensional (1D) systems where the gap is originated by a lattice dimerization (Peierls or SSH effect) as in polyacethylene, as well as low symmetry 1D semiconductors. Results may be applied to several types of conjugated polymers, to semiconducting nanotubes, and to quantum wires of semiconductors. The approach may be generalized to tunneling in strongly correlated systems showing the pseudogap effect, as in the family of high-$T_c$ materials in the undoped limit. We demonstrate the evolution of tunneling current-voltage characteristics from smearing the free electron gap down to threshold for tunneling of polarons and further down to the region of bielectronic tunneling via bipolarons or kink pairs. The interchain tunneling is described in a parallel comparison with the on chain optical absorption, also within the subgap region.

Figure 1

The logarithmic plot for tunneling or absorption intensities $\ln I\sim -S$ in the pseudogap regime: between $2W_1$ and $2{\Delta }_0$.

Figure 2

The logarithmic plot for bielectron tunneling or for optical absorption intensities $\ln I\sim -S$ in the two particle pseudogap regime: between $2W_s$ and $2{\Delta }_0$. Notice the $\sqrt{\Omega -2E_s}$ singularity near the two soliton threshold.

Figure 3

Exact shapes $\Delta (x,E)$ of the equilibrium polaron $E=2^{-1∕2}$ (upper thick line) and of a nearly formed $(E=0.01)$ pair of solitons (lower thick line). Thin lines show exact shapes of optimal fluctuations necessary to create these states by tunneling.

Figure 4

Total energy of the single particle branch $V_1\left(E\right)$ as a function of the energy $E$ of the associated bound state.

Figure 5

(Color online) Total energies $V_{\nu }-\nu U$ for different tunneling branches $V_{\nu }-\nu U$ as a function of the energy $E$ of the associated bound state. This figure corresponds to $U=1.2$ which is below the bielectronic threshold.

Figure 6

(Color online) Tunneling branches $V_{\nu }-\nu U$ as a function of the energy $E$ of the associated bound state. This figure corresponds to $U=1.4$ which is between the thresholds for tunneling of bipolarons and polarons.

Figure 7

(Color online) Tunneling branches $V_{\nu }-\nu U$ as a function of the energy $E$ of the associated bound state (over a selected interval). This figure corresponds to $U=1.8$ which is just at the polaronic threshold, above the bipolaronic one.

Figure 8

(Color online) Tunneling branches $V_{\nu }-\nu U$ as a function of the energy $E$ of the associated bound state. This figure corresponds to $U=1.6$ which is between the thresholds for tunneling of bipolarons and polarons. Contrarily Fig. 6, the Coulomb interaction is taken into account which lifts the crossing degeneracy.