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arXiv:2211.11632 · Physical review. B./Physical review. B

Stability conditions for a large anharmonic bipolaron

Matthew Houtput, Jacques Tempere

Published Jan 20, 2023• 0 citations
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Abstract

A large polaron is a quasiparticle that consists of a nearly free electron interacting with the phonons of a material, whose lattice parameters are much smaller than the polaron scale. The electron-phonon interaction also leads to an attractive interaction between electrons, which can allow two polarons to pair up and form a bipolaron. It has been shown that large bipolarons can form in theory due to strong one-electron--one-phonon coupling, but they have not been seen in real materials because the critical value of the required electron-phonon interaction is too large. Here, we investigate the effect of one-electron--two-phonon coupling on the large bipolaron problem. Starting from a generalization of the Fr\"ohlich Hamiltonian that includes both the standard one-electron--one-phonon interaction as well as an anharmonic one-electron--two-phonon interaction, we use the path-integral method to find a semianalytical upper bound for the bipolaron energy that is valid at all values of the Fr\"ohlich coupling strength $\ensuremath{\alpha}$. We find the bipolaron phase diagram and conditions for the bipolaron stability by comparing the bipolaron energy to the energy of two free polarons. The critical value of the Fr\"ohlich coupling strength ${\ensuremath{\alpha}}_{\text{crit}}$ is calculated as a function of the strength of the one-electron--two-phonon interaction. The results suggest that large bipolaron formation is more likely in materials with significant one-electron--two-phonon interaction as well as strong one-electron--one-phonon interaction, such as strontium titanate.

DOI
10.1103/physrevb.107.014510

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A large polaron is a quasiparticle that consists of a nearly free electron interacting with the phonons of a material, whose lattice parameters are much smaller than the polaron scale. The electron-phonon interaction also leads to an attractive interaction between electrons, which can allow two polarons to pair up and form a bipolaron. It has been shown that large bipolarons can form in theory due to strong one-electron--one-phonon coupling, but they have not been seen in real materials because the critical value of the required electron-phonon interaction is too large. Here, we investigate the effect of one-electron--two-phonon coupling on the large bipolaron problem. Starting from a generalization of the Fr"ohlich Hamiltonian that includes both the standard one-electron--one-phonon interaction as well as an anharmonic one-electron--two-phonon interaction, we use the path-integral method to find a semianalytical upper bound for the bipolaron energy that is valid at all values of the Fr"ohlich coupling strength \ensuremathα\ensuremath{\alpha}\ensuremathα. We find the bipolaron phase diagram and conditions for the bipolaron stability by comparing the bipolaron energy to the energy of two free polarons. The critical value of the Fr"ohlich coupling strength \ensuremathαcrit{\ensuremath{\alpha}}_{\text{crit}}\ensuremathαcrit​ is calculated as a function of the strength of the one-electron--two-phonon interaction. The results suggest that large bipolaron formation is more likely in materials with significant one-electron--two-phonon interaction as well as strong one-electron--one-phonon interaction, such as strontium titanate.
The stability of large Fr\"ohlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$) favors bipolaron formation: (i) The critical electron-phonon coupling parameter ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$ (above which the bipolaron is stable) decreases with increasing ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ and (ii) the critical Coulomb repulsion strength UC{\mathit{U}}_{\mathit{C}}UC​ (below which the bipolaron is stable) increases with increasing ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. The binding energy and the corresponding variational parameters are calculated as a function of \ensuremath{\alpha}, U, and ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (\ensuremath{\alpha}\ensuremath{\gg}1) we obtain for ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\ll}1 that Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​\ensuremath{\approxeq}Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0)+c(u)${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{4}$ with c(u) an explicitly calculated constant, dependent on the ratio u=U/\ensuremath{\alpha}, where U is the strength of the Coulomb repulsion. This relation applies both in two dimensions (2D) and in 3D, but with a different expression for c(u). For ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\gg}${\mathrm{\ensuremath{\alpha}}}^{2}$\ensuremath{\gg}1 we find in 3D Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-${\mathrm{\ensuremath{\alpha}}}^{2}$A(u)ln2{\mathrm{ln}}^{2}ln2(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{2}$) [also with an explicit analytical expression for A(u)] whereas in 2D Eestim2D{\mathit{E}}_{\mathrm{estim}}^{2\mathrm{D}}Eestim2D​\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-\ensuremath{\alpha}\ensuremath{\surd}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\pi}(u-2-\ensuremath{\surd}2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined. \textcopyright{} 1996 The American Physical Society.
In this paper we have studied the properties of a two-fermion system interacting with the phonon field in the framework of the Holstein Hubbard model in one, two and three dimensions. We have chosen the modified Lang-Firsov variational wave function with on-site and nearest-neighbour lattice distortion for the phonon subsystem to obtain an effective electronic Hamiltonian. This effective electronic Hamiltonian is then solved exactly to obtain the binding energy and effective mass of the bipolaron. We investigated the stability of the bipolaronic phase by comparing the ground-state energy of a bipolaron and two free polarons. We observed a critical repulsive on-site Hubbard interaction U c below which the bipolaronic phases are stable for a fixed electron-phonon coupling g . In the absence of on-site repulsion, bipolaronic phases are stable over the entire range of electron-phonon coupling for one dimension, whereas there is a critical electron-phonon coupling g c for formation of a stable bipolaron in two and three dimensions.
The energies of bipolaron states in three- and two-dimensional systems are calculated variationally, treating the electron-phonon interaction in the Fr\"ohlich approximation, and separating the relative motion from the center-of-mass motion. The bipolaron is bound if the electron-phonon coupling constant \ensuremath{\alpha} is larger than 6 in three dimensions and 2 in two dimensions, provided the ratio \ensuremath{\eta}=${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$/${\mathrm{\ensuremath{\epsilon}}}_{0}$ is smaller than a critical value which depends on \ensuremath{\alpha}. The theory yields the free-polaron energies as given in the Lee, Low, and Pines approximation for arbitrary values of the electron-phonon coupling constant \ensuremath{\alpha}. For \ensuremath{\alpha} larger than \ensuremath{\sim}10 and \ensuremath{\sim}4 in the three- and two-dimensional cases, respectively, the lowest free-polaron self-energies are obtained in the strong-coupling approach; it is shown that a trial harmonic envelope wave function, localized both in the center-of-mass and relative coordinates, with a coherent phonon state gives a bound localized bipolaron state of lower energy. It is interesting for application to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductivity that the bipolaron bounds more easily in two dimensions than in three, and that the mean value of the pair radius is a few angstroms. Furthermore, bipolaron states obtained when the linear total momentum is conserved have intrinsically high mobility, which is also an important condition to make the bipolaron mechanism consistent with high-temperature superconductivity.
The case of polaron and bipolaron formation due to electron-phonon coupling was studied on a lattice with soft non-linear intersite interactions. The ground state of a system that consists of one or two electrons coupled to a non-linear lattice was obtained numerically. The binding energy of the bipolaron was examined as a function of the degree of the non-linearity of the lattice. A model was investigated where the interatomic interactions in the lattice are described by a double-well potential. This type of non-linear interaction strongly influences the bipolaron formation and generally improves its stability and increases its binding energy by permitting a local structural change of the lattice.
We present a diagrammatic Monte Carlo study of the properties of the Hubbard-Holstein bipolaron on a two-dimensional square lattice. With a small Coulomb repulsion UUU and with increasing electron-phonon interaction, and when reaching a value about two times smaller than the one corresponding to the transition of light polaron to heavy polaron, the system suffers a sharp transition from a state formed by two weakly bound light polarons to a heavy, strongly bound on-site bipolaron. Aside from this rather conventional bipolaron a new bipolaron state is found for large UUU at intermediate and large electron-phonon coupling, corresponding to two polarons bound on nearest-neighbor sites. We discuss both the properties of the different bipolaron states and the transition from one state to another. We present a phase diagram in parameter space defined by the electron-phonon coupling and UUU. Our numerical method does not use any artificial approximation and can be easily modified to other bipolaron models with longer range electron-phonon and∕or electron-electron interaction.
The Holstein Hubbard and Holstein t-J models are studied for a wide range of phonon frequencies, electron-electron, and electron-phonon interaction strengths on finite lattices with up to ten sites by means of direct Lanczos diagonalization. Previously the necessary truncation of the phononic Hilbert space caused serious limitations to either very small systems (four or even two sites) or to weak electron-phonon coupling, in particular in the adiabatic regime. Using parallel computers we were able to investigate the transition from ``large'' to ``small'' polarons in detail. By resolving the low-lying eigenstates of the Hamiltonian and by calculating the spectral function, we can identify a polaron band in the strong-coupling case, whose dispersion deviates from the free-particle dispersion at low and intermediate phonon frequencies. For two electrons (holes) we establish the existence of bipolaronic states and discuss the formation of a bipolaron band. For the two-dimensional Holstein t-J model, we demonstrate that the formation of hole polarons is favored by strong Coulomb correlations. Analyzing hole-hole correlation functions, we find that hole binding is enhanced as a dynamical effect of the electron-phonon interaction. \textcopyright{} 1996 The American Physical Society.