Lattice Polaron

Abstract

When the coupling with phonons increases, the polaron radius decreases and becomes of the order of the lattice constant. Then, all momenta of the Brillouin zone contribute to the polaron wave function and the effective mass approximation cannot be applied. This regime occurs if the characteristic potential energy E _p (polaron level shift) due to the local lattice deformation is compared or larger than the half-bandwidth D. The strong-coupling regime with the dimensionless coupling constant

$$\lambda \equiv \frac{E_{{\rm p}}}{D} \geq 1$$

(3.1)

is called the small or lattice polaron. In general, E _p is expressed as

$$E_{{\rm p}} = \frac{1}{2N} \sum\limits_{{\rm q}}|\gamma({\rm q})|^{2}\omega_{{\rm q}}$$

(3.2)

for any type of phonons involved in the polaron cloud. For the Fröhlich interaction with optical phonons, one obtains $E_{{\rm p}} \simeq q_{\rm d}e^{2}/\pi k$, where q _d is the Debye momentum [59]. For example, with parameters appropriate for high T _c copper oxides $\epsilon_0 \gg \epsilon \simeq 5$ and $q_{\rm D} \simeq 0.7 {\rm {\AA}}^{-1}$, one obtains $E_{{\rm p}} \simeq 0.6 {\rm eV}$ [123, 124]. The exact value of λ_c when the continuum (large) polaron transforms into the small one depends on the lattice structure, phonon frequency dispersions, and the radius of the electron–phonon interaction, but in most cases the transformation occurs around $\lambda_{{\rm c}} \simeq 1$ [125]. Lattice polarons are expected to be the carriers in oxides, which are strongly polarizable doped semiconductors, if the bare-electron band is narrow enough [26], and in molecular nanowires (Sect. 6.3.2).