Wigner transport models of the electron-phonon kinetics in quantum wires

Two quantum-kinetic models of ultrafast electron transport in quantum wires are derived from the generalized electron-phonon Wigner equation. The various assumptions and approximations allowing one to find closed equations for the reduced electron Wigner function are discussed with an emphasis on their physical relevance. The models correspond to the Levinson and Barker-Ferry equations, now generalized to account for a space-dependent evolution. They are applied to study the quantum effects in the dynamics of an initial packet of highly nonequilibrium carriers, locally generated in the wire. The properties of the two model equations are compared and analyzed.

Figure 1

Levinson transitions link diagonal with FOD elements. The FOD element is presented by the open circle, and $n$ and $n$ stand for the left and right phonon sets $\left\{n_q\right\}$ and $\left\{n_q\right\}$.

Figure 2

The Barker-Ferry counterpart of the first Levinson transition shown on Fig. 1. Here $\pm 1^{″}$ denotes an increase or decrease by unity of the phonons in mode ${\mathbf{q}}^{″}$ in the left or right basis.

Figure 3

Initial peak (dashed line) and the $50\text{−}\mathrm{fs}$ wave vector density presented in a window of positive $k_z$.

Figure 4

Time evolution of the wave vector distribution obtained from the Levinson model. The $175\text{−}\mathrm{fs}$ curve is obtained by direct evaluation of $f(k_z,t)$. The distributions at earlier times are computed indirectly, via the Wigner function.

Figure 5

The $175\text{−}\mathrm{fs}$ wave vector distribution obtained from the Levinson model by direct (L) and indirect (LW) computations via the Wigner function. The difference between the two results is significant; the LW curve becomes negative in the valley between the peaks. The Barker-Ferry curve (solid line) is obtained by direct computations.

Figure 6

(Color online) The Wigner function after $175\mathrm{fs}$ of evolution. The two main peaks are truncated for better resolution. The secondary peaks are formed by electrons which transferred part of their energy to the phonons. The classically forbidden regions are placed on the opposite side of the main peaks, in the nearest left and far right corners of the picture. The regions are populated with rapidly moving electrons.

Figure 7

Electron densities at time 0 and $50\mathrm{fs}$. The ballistic curve (dotted line) coincides everywhere with the $50\text{−}\mathrm{fs}$ solution apart from the central region. There reside electrons slowed down by the scattering events.

Figure 8

Electron density after $175\mathrm{fs}$ evolution. The fastest classical electrons form the fronts of the two ballistic peaks (solid line) slightly above (below) $200\mathrm{nm}$ $(-200\mathrm{nm})$. The fastest quantum electrons of the Levinson (bold line) and Barker-Ferry (dashed line) models reach distances placed farther away from the origin.

Figure 9

Ballistic (solid line), Levinson (bold line), and Barker-Ferry (dashed line) distributions of the mean energy per particle at $175\mathrm{fs}$.