Polaron Theory in Wide and Narrow Electron Bands

The polaron band theory of Lee, Low and Pines is extended beyond the effective mass approximation to a general form of the electron–phonon interaction within two different self‐consistent schemes. In the first one a variational Bloch state depending on the polaron crystalline momentum is used to study polaron features taking into account umklapp processes. In particular, the self‐energy and the effective mass at the Γ point of the Brillouin zone are calculated as functions of the electron–phonon coupling constant α. It is found that a sudden transition of these quantities from a large band to a narrow band polaron behaviour occurs at a critical value α c . Such value increases indefinitely with the bare electron band width; moreover the transition becomes sharper. The theory recovers, as particular cases, a) the Lee‐Low‐Pines regime valid for small α and large electron band width Δ˜(Δ˜ ≫ ħω and b) the Holstein limit for large α and small band width (Δ˜ ≫ ħω). This occurs as result of the competing role of α and Δ˜. In the second scheme, the electron–phonon interaction is considered only in a single elementary cell and the self‐trapped polaron state is calculated self‐consistently. It is shown that, if only one electronic band is taken into account, such state becomes a Δ function with infinitely negative energy when α is larger than a critical value depending on the electron band width. To avoid this fact, electronic band states of higher energy must be considered. Finally, through the tight‐binding method, the polaron band is set up. It is shown that at the Γ point of the Brillouin zone, the contribution of the higher electronic bands prevents the self‐energy and the polaron mass to have a sharp increase from the large to the narrow band regime.