Scattering of an electronic wave packet by a one-dimensional electron-phonon-coupled structure

We investigate the scattering of an electron by phonons in a small structure between two one-dimensional tight-binding leads. This model mimics the quantum electron transport through atomic wires or molecular junctions coupled to metallic leads. The electron-phonon-coupled structure is represented by the Holstein model. We observe permanent energy transfer from the electron to the phonon system (dissipation), transient self-trapping of the electron in the electron-phonon-coupled structure (due to polaron formation and multiple reflections at the structure edges), and transmission resonances that depend strongly on the strength of the electron-phonon coupling and the adiabaticity ratio. A recently developed TEBD algorithm, optimized for bosonic degrees of freedom, is used to simulate the quantum dynamics of a wave packet launched against the electron-phonon-coupled structure. Exact results are calculated for a single electron-phonon site using scattering theory and analytical approximations are obtained for limiting cases.

Figure 1

The electronic density distribution calculated with TEBD-LBO for a six-site EPC wire with phonon frequency ${\omega }_0=2.25t_0/\hslash$ and electron-phonon coupling $\gamma =t_0$ at three instances of time: before (red solid line), during (black dots), and after (blue dashed line) the main scattering processes. The index $1\le j\le L=560$ numbers the lattice sites. The red solid and blue dashed curves are multiplied by factors 2 and 10, respectively. The inset shows an enlarged view of the region around the EPC wire. Thin vertical lines show the position of the first and last EPC sites.

Figure 2

Schematic representation of the matrix-product state used by the TEBD-LBO algorithm. The $\mathrm{\Gamma }$ matrices and the diagonal $\lambda$ matrices have dimensions smaller or equal to $D$ and have the same properties as in the original TEBD algorithm [23]. The matrix $R$ describes the local transformation from the $d$-dimensional local bare basis to the $d_O$-dimensional optimized subbasis.

Figure 3

Time evolution of the numerical transmission coefficient (19) calculated with TEBD-LBO for various cases: free particle (red solid line), EPC impurity $\left(L_H=1\right)$ in the adiabatic strong-coupling regime with $\hslash {\omega }_0=0.2t_0$ and $\gamma =1.9t_0$ (blue dotted line) as well as with $\hslash {\omega }_0=0.6t_0$ and $\gamma =3.9t_0$ (purple dashed line), and EPC structure with one site (black dashed-dotted line) and three sites (gray double-dashed line) in the intermediate regime $(\hslash {\omega }_0=1.6t_0$ and $\gamma =1.85t_0)$. The center of the wave packet reaches the first EPC site at time $t\approx (L_{\text{TB}}-j_0)a/v\approx 25\hslash /t_0$ in all cases.

Figure 4

(a) Transmission coefficients calculated using TEBD-LBO (lines) and with scattering theory and wave packet averaging (symbols) for an EPC impurity $\left(L_H=1\right)$. Results are plotted as a function of the electron-phonon coupling $\gamma$ for the adiabatic regime $\hslash {\omega }_0=0.4t_0$ (gray solid line and triangles), the intermediate regime $\hslash {\omega }_0=1.35t_0$ (blue dotted line and squares), and the anti-adiabatic regime $\hslash {\omega }_0=10t_0$ (red dashed line and dots). For $\hslash {\omega }_0=10t_0,\gamma$ is divided by a factor 10. (b) Components ${\left|\langle n|\psi \rangle \right|}^2$ of the occupied-site optimal mode $|\psi \rangle$ in the bare boson basis $|n\rangle$ calculated with TEBD-LBO for $\hslash {\omega }_0=10t_0$ and $\gamma =\hslash {\omega }_0$ (red solid line) as well as $\gamma =\sqrt{2}\hslash {\omega }_0$ (blue dashed line). Also shown are the first (circles) and second (triangles) excited states of the single-site Holstein model occupied by one electron.

Figure 5

(a) Transmission coefficients calculated with TEBD-LBO in the anti-adiabatic regime $\left(\hslash {\omega }_0=10t_0\right)$ as a function of the electron-phonon coupling $\gamma$ for several EPC wire lengths: $L_H=1$ (red dashed-dotted line), $L_H=2$ (blue dashed line), $L_H=3$ (black dotted line), and $L_H=4$ (gray solid line). (b) Components ${\left|\langle n|\psi \rangle \right|}^2$ of the two most important optimal modes $|\psi \rangle$ for a site occupied by an electron in the bare boson basis $|n\rangle$ calculated with TEBD-LBO for $L_H=2$ and $\hslash {\omega }_0=\gamma =10t_0$ at two points in time: First (red solid line) and second (orange dashed line) optimal modes at $t=25\hslash /t_0$ and first (gray dotted line) and second (blue dashed-dotted line) optimal modes at $t=75\hslash /t_0$. Also shown are the ground state (circles) and first excited state (squares) of the single-site Holstein model occupied by one electron.

Figure 6

Reflection coefficient $R\left(L_H\right)$ calculated with TEBD-LBO as a function of the electron-phonon coupling $\gamma$ in the weak-coupling adiabatic regime $\left(\hslash {\omega }_0=0.6t_0\right)$ for EPC wires of length $L_H=2$ (crosses), $L_H=6$ (squares), and $L_H=11$ (circles). The lines show the predictions of scattering theory for an EPC impurity combined with Eq. (21) for $R\left(2\right)$ (red solid line), $R\left(6\right)$ (blue dotted line), and $R\left(11\right)$ (purple dashed line).

Figure 7

Phonon energy calculated with TEBD-LBO as a function of time. (a) For an EPC impurity with $\hslash {\omega }_0=1.35t_0$ and three electron-phonon couplings $\gamma =0.8t_0$ (red solid line), $\gamma =1.45t_0$ (blue dotted line), and $\gamma =1.8t_0$ (black dashed line). (b) For a four-site EPC structure in the intermediate regime $\left(\hslash {\omega }_0=1.6t_0,\gamma =1.85t_0\right)$ the phonon energies are shown separately for the first (red crosses), second (blue squares), third (black bullets), and fourth (gray triangles) site.

Figure 8

Dissipated energies calculated using TEBD-LBO (symbols) and with scattering theory and wave packet averaging (lines) for an EPC impurity $\left(L_H=1\right)$. Results are plotted as a function of the electron-phonon coupling $\gamma$ for the adiabatic regime $\hslash {\omega }_0=0.4t_0$ (gray solid line and triangles), the intermediate regime $\hslash {\omega }_0=0.9t_0$ (blue dotted line and squares), and close to the limit of the nondissipative regime $\hslash {\omega }_0=1.5t_0$ (red dashed line and dots).

Figure 9

(a) Electronic density on the EPC impurity (red solid line) and phonon energy (blue dashed line) calculated with TEBD-LBO as a function of time for $L_H=1,\hslash {\omega }_0=0.20t_0$, and $\gamma =1.90t_0$. The electron density is multiplied by a factor 10. The black dotted line shows Eq. (25) with $t_{\text{off}}=24t_0/\hslash ,E_{\text{off}}=0.12t_0$, and a constant average density $n_j(t=40\hslash /t_0)=0.0116$. (b) Electronic densities (multiplied by a factor 100) calculated with TEBD-LBO as a function of time on the left (red line) and right (blue dots) sites of a two-site EPC structure with $\hslash {\omega }_0=\gamma =10t_0$.

Figure 10

The rightmost part of the electronic density distributions $n_j\left(t\right)$ calculated with TEBD-LBO at $t=60\hslash /t_0$ as a function of the lattice site index $j$ for $L_H=1,\hslash {\omega }_0=1.35t_0$ and several electron-phonon couplings corresponding to transmission minima at $\gamma =1.5t_0$ (blue dotted line) and at $\gamma =2.45t_0$ (black dashed-dotted line) as well as to transmission maxima at $\gamma =1.80t_0$ (orange dashed line) and at $\gamma =2.65t_0$ (gray double-dashed line). The vertical black line shows the position of the maximum of the free Gaussian wave packet (red line).