Polarons from first principles

TL;DR

The article surveys a broad spectrum of ab initio polaron theories, linking historical effective-Hamiltonian models to modern first-principles methods. It presents a unifying Hedin-Baym framework that encompasses canonical transformations, Green's functions, and DFT-based polaron equations, clarifying how polaron energies, wavefunctions, and lattice distortions are computed from first principles. The review highlights advances in polaron spectral functions, mobility, exciton polarons, and real materials, while detailing key challenges such as anharmonicity, temperature, and finite-density effects. It also emphasizes how these methods converge to provide predictive insights into polaron behavior across alkali halides, oxides, halide perovskites, and 2D materials, with implications for transport and optoelectronic applications.

Abstract

This article reviews recent theoretical developments in the ab initio study of polarons in materials. The polaron is an emergent quasiparticle that arises from the interaction between electrons and phonons in solids, and consists of an electron or a hole accompanied by a distortion of the crystal lattice. Recent advances in experiments, theory, and computation have made it possible to investigate these quasiparticles with unprecedented detail, reigniting the interest in this classic problem of condensed matter physics. Recent theoretical and computational advances include ab initio calculations of polaron spectral functions, wavefunctions, lattice distortions, and transport and optical properties. These developments provide new insight into polaron physics, but they have evolved somewhat independently from the earlier effective Hamiltonian approaches that laid the foundation of the field. This article aims to bridge these complementary perspectives by placing them within a single unified conceptual framework. To this end, we start by reviewing effective Hamiltonians of historical significance in polaron theory, ab initio techniques based on density functional theory, and many-body first-principles approaches to polarons. After this survey, we outline a general field-theoretic framework that bridges between these diverse approaches to polaron physics. For completeness, we also review recent progress in the study of exciton polarons and self-trapped excitons and their relations to polarons. Beyond the methodology, we discuss recent applications to several classes of materials that attracted attention in the context of polaron physics.

Figures (26)

• Figure 1: Illustration of small polarons (a) and large polarons (b) in rutile TiO2, as obtained from first-principles calculations. The isosurfaces represent the square modulus of the polaron wavefunction, the green arrows in (a) represent the accompanying lattice distortion. Ti and O are in blue and red, respectively. From Dai_Giustino_2024c.
• Figure 2: (a) Effective potential energy surface of the Landau-Pekar polaron, Eq. \ref{['Eq:LP_E_vs_rp']}. The black curve is $E(r_{\rm p})$, the blue curve is the attractive potential energy term, the red curve is the repulsive kinetic energy term. (b) Radius of the Landau-Pekar polaron (black line), from Eq. \ref{['Eq:r_LP_alpha']}. For comparison, we also show the radius of the Feynman model (Sec. \ref{['Sec:Feynman']}) in the small-$\alpha$ limit (blue line), from Eq. \ref{['Eq:radius_feynm1']}. At large $\alpha$, the Feynman result (not shown) coincides with the Landau-Pekar result.
• Figure 3: Formation energy of the Fröhlich polaron as a function of the electron-phonon coupling strength $\alpha$. The energy zero corresponds to a free electron in the absence of lattice distortion. The black line is the Landau-Pekar solution, Eq. \ref{['Eq:Etot_LP_alpha']}, while the blue line and red line are the asymptotic expansions of Feynman's solution at weak coupling and strong coupling, respectively, Eqs. \ref{['Eq:Emin_feynm_alpha1']}-\ref{['Eq:Emin_feynm_alpha2']}. The weak-coupling expansion coincides with Fröhlich's and LLP's solution, the strong-coupling expansion coincides with the Landau-Pekar solution apart from a constant shift. Black circles are DMC data from Hahn_Franchini_2018. The regions of weak, intermediate, and strong coupling are marked qualitatively.
• Figure 4: Formation energy of the Holstein polaron in one dimension, as a function of the Holstein electron-phonon coupling strength $\lambda$ and for two different values of the "adiabaticity parameter" $\hbar\omega/t$. The energy zero corresponds to a band electron without lattice distortion. The black lines correspond to the weak coupling and strong coupling expressions in Eqs. \ref{['Eq:Holstein_weak']} and \ref{['Eq:Holstein_strong']}, respectively. The disks are DMC data from Macridin_Sawatzky_2003 (Macridin_Sawatzky_2003, Fig. 3.6), for the adiabaticity parameters $\hbar\omega/t = 0.1$ (red) and 0.5 (blue). The color shading marks qualitatively the weak-, intermediate-, and strong-coupling regimes.
• Figure 5: Schematic representation of the energies involved in the polaron formation process in the dilute limit, i.e. infinitely large supercells. The solid blue line represents the $N$-electron system, and the solid black line represents the $(N + 1)$-electron system. The dashed green line represents a fictitious $(N + 1)$-electron system in which an electron has been added to the conduction band minimum ($\varepsilon_{\rm CBM}$). The excitation energy from the $(N + 1)$-electron ground state (at $\boldsymbol{\tau}_0 + \Delta \boldsymbol{\tau}$) to the distorted $N$-electron state is the electron addition or removal energy $\varepsilon_{\rm p}$. The polaron formation energy, that is the energy gained by the system when a delocalized electronic state becomes localized in a polaronic state, is represented by $\Delta E_\mathrm{f}$.