Many-body Green's function approaches to the doped Fröhlich solid: Exact solutions and anomalous mass enhancement

In polar semiconductors and insulators, the Fröhlich interaction between electrons and long-wavelength longitudinal optical phonons induces a many-body renormalization of the carrier effective masses and the appearance of characteristic phonon sidebands in the spectral function, commonly dubbed “polaron satellites.” The simplest model that captures these effects is the Fröhlich model, whereby electrons in a parabolic band interact with a dispersionless longitudinal optical phonon. The Fröhlich model has been employed in a number of seminal papers, from early perturbation-theory approaches to modern diagrammatic Monte Carlo calculations. One limitation of this model is that it focuses on undoped systems, thus ignoring carrier screening and Pauli blocking effects that are present in real experiments on doped samples. To overcome this limitation, we here extend the Fröhlich model to the case of doped systems, and we provide exact solutions for the electron spectral function, mass enhancement, and polaron satellites. We perform the analysis using two approaches, namely, Dyson's equation with the Fan-Migdal self-energy, and the second-order cumulant expansion. We find that these two approaches provide qualitatively different results. In particular, Dyson's approach yields better quasiparticle masses and worse satellites, while the cumulant approach provides better satellite structures, at the price of worse quasiparticle masses. Both approaches yield an anomalous enhancement of the electron effective mass at finite doping levels, which in turn leads to a breakdown of the quasiparticle picture in a significant portion of the phase diagram.

Figure 1

Schematic illustration of the many-body renormalization of the conduction band bottom of a polar semiconductor or insulator by the Fröhlich interaction. The illustration refers to a parabolic conduction band minimum, doped with electrons up to the Fermi level. The dashed line indicates the noninteracting band structure, and the solid lines show the renormalized band minimum as well as the phonon sidebands (two sidebands, for example). The energy separation between the quasiparticle band and the sidebands is an integer multiple of the LO phonon energy.

Figure 2

Self-energy and spectral function for the undoped, empty-band Fröhlich model with $\alpha =1$. (a) Real part of the greater self-energy (black lines) relative to the dispersion of the noninteracting electron in units of the phonon energy $\hslash {\omega }_0$ (dashed orange line). The blue area indicates the energy range $[{\epsilon }_0-\hslash {\omega }_0,{\epsilon }_0+\hslash {\omega }_0]$. (b) Imaginary part of the greater self-energy. (c) Color plot of the Dyson spectral function of the undoped system. The dashed orange line indicates the dispersion of the noninteracting electron in units of the phonon energy $\hslash {\omega }_0$. (d) Logarithmic line plot of the Dyson spectral function at the $\mathrm{\Gamma }$ point. (e) Color plot of the second-order cumulant spectral function. (f) Logarithmic line plot of the cumulant spectral function at the $\mathrm{\Gamma }$ point.

Figure 3

(a) Electron energy renormalization in the undoped Fröhlich model as a function of the coupling strength $\alpha$. Shown in blue are the weak- and strong-coupling limits of the Feynman model. (b) Electron mass renormalization in the undoped Fröhlich model as a function of the coupling strength $\alpha$.

Figure 4

Self-energy and spectral function for the Fröhlich model with $\alpha =1$ and a Fermi energy of $E_{\mathrm{F}}/\hslash {\omega }_0=0.4$. Free-carrier screening is not included. (a) Real part of the lesser and greater self-energies in the extreme antiadiabatic limit (black lines) relative to the dispersion of the noninteracting particle (dashed orange line). The solid and dashed red lines indicate the Fermi energy and Fermi momentum, respectively; the blue area indicates the energy range $[E_{\mathrm{F}}-\hslash {\omega }_0,E_{\mathrm{F}}+\hslash {\omega }_0]$. (b) Imaginary part of the self-energy. (c) Color plot of the Dyson spectral function in the extreme antiadiabatic limit relative to the noninteracting electron energy (dashed orange line) and the Fermi energy (red line). (d) Logarithmic line plot of the Dyson spectral function at the $\mathrm{\Gamma }$ point. (e) Color plot of the second-order cumulant spectral function. (f) Logarithmic line plot of the cumulant spectrum at $k=0$.

Figure 5

(a) Renormalized Dyson QP energy $E_0/\hslash {\omega }_0$ relative to the Fermi level. Negative QP energies $E_0$ (shown in green) indicate a higher binding energy of the interacting system, whereas positive energies (shown in red) imply that the renormalized QP energy lies above the Fermi level, and indicate a breakdown of the Fermi surface. (b) Renormalized Dyson effective mass at the $\mathrm{\Gamma }$ point. Positive values of $m^{*}$ (shown in green) indicate the renormalization of the QP mass; negative values of $m^{*}$ (shown in red) indicate that the curvature of the QP spectrum at $k=0$ has become negative, and signal a breakdown of the second-order expansion of the self-energy. The inset shows an enlarged view for $\alpha <1$ and $E_{\mathrm{F}}<0.2\hslash {\omega }_0$; note that the color scale in the inset has been extended to $m^{*}/m_0=10$. (c) Renormalized second-order cumulant QP energy $E_0/\hslash {\omega }_0$ relative to the Fermi level. (d) Renormalized second-order cumulant effective mass.

Figure 6

(a) Imaginary lesser self-energy at ${\epsilon }_k=0.1\hslash {\omega }_0$ at a Fermi energy of $E_{\mathrm{F}}=0.4\hslash {\omega }_0$ and $\alpha =1$ (black line). Contributions to Eq. (39) arising from the singularity in the noninteracting electron Green's function are shown in green; those due to phonon emission and absorption processes across the entire Fermi sea appear in blue. (b) Corresponding satellite function $A^{\text{S}}\left(\omega \right)$ (before convolution with the QP function, shown in red). Contributions to the satellite function due to the electron Green's function and phonon emission and absorption processes are shown in green and blue, respectively.

Figure 7

Effect of free-carrier screening on the electron-phonon coupling matrix element $g$ for a range of typical doping values for semiconductors. (a) Screening function $|{\varepsilon }^{\text{RPA}}{|}^{-2}$ vs wavenumber $q$, evaluated at the phonon frequency ${\omega }_0$ for a dilute electron gas. The parameters ${\varepsilon }_0$ and $m_0$ given in the legend correspond to ${\mathrm{SrTiO}}_3$. The dashed line at $|{\varepsilon }^{\text{RPA}}{|}^{-2}=1$ indicates the limit of no free-carrier screening. (b) Same as (a), but for a dense electron gas. The parameters ${\varepsilon }_{\infty }$ and $m_0$ given in the legend correspond to GaAs.

Figure 8

Self-energy for the Fröhlich model with $\alpha =1$ and a Fermi energy of $E_{\mathrm{F}}/\hslash {\omega }_0=0.8$ without free-carrier screening $\left[\mathrm{\Sigma }\right(\omega \left)\right]$ and with free-carrier screening effects $\left[\mathrm{\Sigma }\right(\omega )+\varepsilon ]$, respectively. (a) Real part of the lesser and greater self-energies without free-carrier screening (black lines) relative to the dispersion of the noninteracting particle (dashed orange line). The solid and dashed red lines indicate the Fermi energy and Fermi momentum, respectively. The blue area indicates the energy range $[E_{\mathrm{F}}-\hslash {\omega }_0,E_{\mathrm{F}}+\hslash {\omega }_0]$. (b) Real part of the lesser and greater self-energies including free-carrier screening, assuming a dilute electron gas like in the conduction band of ${\mathrm{SrTiO}}_3$. (c) Imaginary part of the self-energy without free-carrier screening. (d) Imaginary part of the self-energy including free-carrier screening.

Figure 9

(a) Unscreened Dyson spectral function at $\alpha =1$ and $E_{\mathrm{F}}/\hslash {\omega }_0=0.8$. The noninteracting electron energy is indicated by the dashed gray line, the Fermi level by the solid red line. (b) Unscreened second-order cumulant spectral function. (c) Logarithmic line plot of the Dyson spectral weight function at $k=0$. (d) Logarithmic line plot of the cumulant spectral function at $k=0$. (e) Screened Dyson spectral function assuming a dilute electron gas like in the conduction band of ${\mathrm{SrTiO}}_3$. (f) Screened cumulant spectral function. (g) Logarithmic line plot of the screened Dyson spectral function. (h) Logarithmic line plot of the screened cumulant spectral function.

Figure 10

(a) Renormalized Dyson quasiparticle energy $E_0/\hslash {\omega }_0$ relative to the Fermi level, including screening by free carriers. Negative QP energies (shown in green) indicate a higher binding energy of the interacting system, whereas positive energies (in red) imply that the renormalized QP energy lies above the Fermi level, and indicate a breakdown of the Fermi surface. (b) Renormalized second-order cumulant quasiparticle energy relative to the Fermi surface. (c) Renormalized Dyson effective mass at the $\mathrm{\Gamma }$ point. Positive values of $m^{*}$ (shown in green) indicate the renormalization of the QP mass; negative values of $m^{*}$ (shown in red) indicate that the curvature of the QP spectrum at $k=0$ has become negative, and signal a breakdown of the second-order expansion of the self-energy. (d) Renormalized second-order cumulant effective mass.