Dynamic electron localization initiated by particle-bath coupling

We consider the quantum evolution and relaxation of an electron or hole which is coupled to a set of bath modes. In most applications the bath modes would be the vibronic coordinates but the model considered applies to any type of dynamic boson environment. The method is developed specifically for the problem of dynamic polaron formation in small nonperiodic systems. It can describe a broad group of experimental situations, including in particular electron localization in organics and polymeric materials and devices. The immediate bath is allowed to dissipate energy to a secondary bath. The bath obeys classical dynamics which puts some restriction on the range of validity of this approach. Using the density matrix formalism on a tight binding model consisting of a linear chain coupled to vibronic coordinates, we demonstrate in real time how the interaction with a dissipative bath makes the initial quantum distribution reach a steady-state population. This calculation is based on the Ehrenfest dynamics approximation. As an example we consider coupling at a single impurity site and find that for given parameters (bath coupling, site energy, and relaxation rate), the particle becomes dynamically localized in space on a particular time scale. This localized particle can be called a polaron. We define a population formation time in the same way as done in the experimental measurement. This formation time is studied as a function of the coupling strength, bandwidth, and energy dissipation rate. Energy dissipation plays a crucial role in the spatial localization process. The formation time shortens as the electron-vibration coupling increases, and as the intersite tunneling increases, but lengthens with impurity trap depth. Polaron formation is suppressed for sufficiently wide electronic bands.

Figure 1

The scheme of a chain including five sites. The electron-vibration (e-vib) interaction is added to the middle site 2.

Figure 2

Population distribution on different sites shown as a function of time. ${\varepsilon }_l=0$ ($l=0,1,3,4$), ${\varepsilon }_2=-0.2\mathrm{eV}$, $V=0.1\mathrm{eV}$, ${\lambda }_0{\tau }_0=0.01\mathrm{eV}$, $\hslash {\omega }_0=0.1\mathrm{eV}$, $\hslash {\gamma }_0=0.04\mathrm{eV}$, ${\tau }_0=\frac{\hslash }{\mathrm{eV}}\approx$ 0.65 fs. In the left top of panel (b), the green and blue lines indicate the average values of $P_0$ and $P_2$, respectively. In the other panels the red line indicates the average value of the oscillation shown in black. No electron-vibration coupling [$\alpha =0$ in Eq. (2)] was used for panel (a) and the dashed line indicates $P_2$.

Figure 3

(a) Stationary population distribution on different sites $l$ with different band coupling $V$. (b) Population ($P_2$) shown as a function of time. ${\varepsilon }_l=0$ ($l=0,1,3,4$), ${\varepsilon }_2=-0.2\mathrm{eV}$, ${\lambda }_0{\tau }_0=0.01\mathrm{eV}$, $\hslash {\omega }_0=0.1\mathrm{eV}$, $\hslash {\gamma }_0=0.04\mathrm{eV}$, ${\tau }_0=\frac{\hslash }{\mathrm{eV}}\approx$ 0.65 fs. In panel (b) the red line indicates the average value of the oscillation shown in black.

Figure 4

Population distribution on site $l$ shown with different total site number $N$. In the $X$ axis, 0 indicates the middle site ($N/2$ with $N$ being an even number). ${\varepsilon }_l=0$ but with the middle site energy ${\varepsilon }_{N/2}=-0.2\mathrm{eV}$, $V=0.1\mathrm{eV}$, $\hslash {\omega }_0=0.1\mathrm{eV}$, $\hslash {\gamma }_0=0.04\mathrm{eV}$, ${\tau }_0=\frac{\hslash }{\mathrm{eV}}\approx$ 0.65 fs.

Figure 5

(a) Population formation time ${\tau }_p$ [Eq. (21)] shown as a function of the electron-vibration coupling parameter ${\lambda }_0$ and vibration-phonon coupling relaxation parameter ${\gamma }_0$. (b) Population formation time ${\tau }_p$ [Eq. (21)] shown as a function of the nearest neighbor site coupling parameter $V$ and ${\lambda }_0$. ${\varepsilon }_l=0$ ($l=0,1,3,4$), ${\varepsilon }_2=-0.2\mathrm{eV}$, $V=0.1\mathrm{eV}$, $\hslash {\omega }_0=0.1\mathrm{eV}$, $\hslash {\gamma }_0=0.04\mathrm{eV}$, ${\tau }_0=\frac{\hslash }{\mathrm{eV}}\approx$ 0.65 fs.

Figure 6

(a) Population $P_2$ shown as a function of time with different electron-vibration coupling parameter ${\lambda }_0$ and the vibration-phonon coupling dissipation rate relaxation parameter ${\gamma }_0=0.04$ eV. (b) Population $P_2$ shown as a function of time with different phonon bath relaxation parameter ${\gamma }_0$ and ${\lambda }_0{\tau }_0=0.01$ eV. ${\varepsilon }_l=0$ ($l=0,1,3,4$), ${\varepsilon }_2=-0.2$ eV, $V=0.1$ eV, $\hslash {\omega }_0=0.1$ eV, ${\tau }_0=\frac{\hslash }{\mathrm{eV}}\approx$ 0.65 fs. The red line indicates the average value of the oscillation shown in black.

Figure 7

Total system energy $E_T$ shown as a function of time with different vibration-phonon coupling dissipation rate ${\gamma }_0$ and $\Delta E=0.2$ eV. ${\varepsilon }_l=0$ ($l=0,1,3,4$), ${\varepsilon }_2=-0.2$ eV, $V=0.1$ eV, ${\lambda }_0{\tau }_0=0.8$ eV, $\hslash {\omega }_0=0.01$ eV, ${\tau }_0=\frac{\hslash }{\mathrm{eV}}\approx$ 0.65 fs.