Interfacial Polaron–Electron Liquid Transition

• The paper demonstrates that interfacial polaron–electron liquid transition drives a quantum phase change, shifting from localized, strongly coupled polarons to a delocalized, coherent electron liquid.
• Experimental studies using ARPES and HREELS reveal distinct spectroscopic signatures, such as band replicas and plasmon mode softening, that confirm the role of electron–phonon coupling.
• Theoretical models, including minimal Hamiltonians and mean-field approaches, quantify critical coupling parameters and phase boundaries that underpin enhanced interfacial superconductivity.

The interfacial polaron–electron-liquid transition is a quantum phase transition that occurs at polar oxide interfaces—such as single-layer FeSe on SrTiO₃ (FeSe/STO) or LaAlO₃/SrTiO₃—where mobile electrons, transferred or accumulated at the interface, undergo a crossover from localized, strongly coupled polaron states to a delocalized, coherent electron liquid. This transition is governed by the interplay between electronic kinetic energy, electron–phonon (E–ph) coupling, long-range Coulomb interaction, and electronic screening, manifesting in distinct spectroscopic and transport signatures. The phenomenon is central to both enhanced interfacial superconductivity and the emergence of complex electronic phases at oxide heterostructures (Zhang et al., 2018, Nanda et al., 2010).

1. Dynamic Interfacial Polarons and Non-Adiabatic Coupling

At interfaces like FeSe/STO, electron transfer from the overlayer (FeSe) to the substrate (STO) induces strong polarization of the STO lattice. The dominant surface phonon modes of STO—Fuchs–Kliewer modes at ℏωβ ≈ 59 meV and ℏωα ≈ 97 meV—have energies comparable to or exceeding the FeSe Fermi energy (E_F ≈ 56 meV). When the phonon period τph ≡ 2π/ωα ≈ 41 fs is much shorter than the electronic momentum relaxation time τ_e ≈ 130 fs, the adiabatic Migdal–Eliashberg approximation fails, and the system enters a non-adiabatic regime. Electrons are “dressed” by dynamic polarization clouds—i.e., time-dependent local lattice distortions—forming dynamic interfacial polarons. These quasiparticles embody the non-trivial entanglement of fast electron motion and instantaneous lattice response (Zhang et al., 2018).

2. Experimental Observations: Spectroscopic Signatures

Experimental identification of the polaron–liquid crossover leverages inelastic electron scattering (HREELS) and angle-resolved photoemission spectroscopy (ARPES):

• HREELS spectra on Nb-doped STO display only the STO surface phonons at 59 and 97 meV and a broad plasmonic mode at ≈180 meV (ρ′). When capped with FeSe, a new polaronic plasmon mode ρ at ≈150 meV appears, with temperature-dependent softening that parallels the evolution of STO phonons, indicating the formation of a new, charge-transfer-renormalized collective excitation (Zhang et al., 2018).
• ARPES measurements reveal band replicas shifted by ≈97 meV, consistent with strong, forward-scattering electron–phonon coupling. The phonon linewidth broadens markedly at the interface (Γ_α(q) peaks at q ≈ 0.2 Å⁻¹), directly evidencing polaron dressing by the α mode.
• Polaron formation is also inferred from charge disproportionation measured in interface models, where a polaronic insulating phase exhibits order-1 differences in site occupancies (Nanda et al., 2010).

3. Theoretical Models and the Phase Boundary

The underlying physics can be described by a minimal model Hamiltonian incorporating electronic motion (tight-binding), phonons (Holstein or Fröhlich type), and E–ph interaction:

$$H = \sum_k \epsilon_k c^\dagger_k c_k + \sum_q \hbar \omega_{LO} b^\dagger_q b_q + \sum_{k,q} g_q c^\dagger_{k+q}c_k(b_q + b^\dagger_{-q})$$

The polaron–electron liquid transition is controlled by:

• Dimensionless electron–phonon coupling: $\lambda \equiv 2E_P/W$ (with $E_P$ the polaron binding energy, $W$ the electronic bandwidth).
• Phonon energy: $\hbar\omega_{LO} \approx 97~\mathrm{meV}$.
• Carrier density: $n$ determines screening and localization.

Critical values are established either by comparing the polaron binding energy to the kinetic energy scale (e.g., $E_P \sim \frac{1}{2}W$) or via explicit mean-field solutions. In FeSe/STO, $\lambda \sim 0.6–0.8$ at optimal doping ($n \approx 10^{14}$ cm⁻²), placing the system near the crossover between localized polarons and a coherent liquid (Zhang et al., 2018). In the Hubbard–Holstein model (e.g., for LAO/STO) the critical coupling is $\lambda_c \approx 0.34$ eV (determined numerically), with polaron formation favored when $\lambda > \lambda_c$ (Nanda et al., 2010).

4. Collective Polaronic Dynamics and Plasmon Modes

The delocalized regime supports collective charge excitations—polaronic plasmons. The long-wavelength plasmon dispersion for interfacial polarons is given by:

$$\omega_p(q) \approx \sqrt{ \frac{n_p e^2 q}{2\epsilon_\mathrm{eff} m_p} }$$

With coupling to the lattice-relevant α-phonon, the system exhibits hybridized modes:

$$\omega_{\pm}(q) = \sqrt{ \frac{1}{2} [\omega_p(q)^2 + \omega_\alpha^2] \pm \frac{1}{2} \sqrt{ [\omega_p(q)^2 - \omega_\alpha^2]^2 + 4g^2 } }$$

Experimentally, the emergent ρ mode sits between the phonon and the bare STO plasmon, softening in tandem with α, demonstrating strong interfacial polaronic coupling (Zhang et al., 2018).

5. Competing Interactions and Phase Separation

At large electron–phonon coupling and strong Coulomb repulsion, the interface may phase separate into polaron-rich (localized, insulating) and electron-rich (delocalized, metallic) domains. Within the Hubbard–Holstein mean-field framework, the critical line for polaron formation is given approximately by:

$\lambda_c(U) = zt + U/2 + e^2/(\epsilon a)$

where $t$ is the nearest-neighbor hopping, $U$ is the Hubbard repulsion, $z$ is the lattice coordination, and $a$ is the lattice constant. The resultant phase diagram includes 2D polaronic insulator, 2D/3D electron liquid, and mixed-phase (nanoscale domains, size $\sim 1$ nm); temperature effects can melt the phase separation above tens of kelvin (Nanda et al., 2010). Nanoscale cohabitation of metallic and insulating regions at the interface is a hallmark outcome when $\lambda > \lambda_c(\epsilon)$.

6. Impact on Interfacial Superconductivity

In the itinerant polaron–electron liquid regime, enhanced superconductivity arises from two principal mechanisms:

1. Short-range polaron–polaron attraction, $V_{pp} \approx 40$ meV, derives from instantaneous α phonon deformation stabilizing Cooper pairs.
2. Hybrid bosonic spectrum: the mixing of polaronic plasmon and STO phonons increases low-energy bosonic phase space available for pairing.

Combined with the inherent FeSe pairing strength ($T_c^0 \approx 40$ K from electron doping), these interfacial effects elevate the superconducting transition temperature up to $T_c \sim 65$ K. Quantitative estimates using a McMillan-type formula with total coupling $\lambda_{tot} = \lambda_\mathrm{FeSe} + \lambda_p$ recover the observed $T_c$ enhancement (Zhang et al., 2018). A plausible implication is that analogous interfacial mechanisms may be leveraged more broadly to engineer high-$T_c$ systems.

7. Characteristic Length and Energy Scales

Physical parameters defining the nature of the transition, as extracted from models and experiment, include:

Quantity	Typical Value (FeSe/STO, LAO/STO)	Context
Polaron radius $R_p$	$R_p \sim t/\lambda$; $R_p \lesssim a \approx 4$ Å	Localized polarons
Effective mass $m^*/m$	$m^*/m \approx \exp(\lambda/\omega_0)$ (Holstein)	Mass renormalization
Sheet density $n_s$	$n_s \approx 1-3 \times 10^{14}$ cm⁻²	Metallic interface
Conductivity $\sigma$	$\sigma \approx 10^3-10^4$ Ω⁻¹·cm⁻¹	Drude metallic state
Domain size $N$	$N \approx 5$ sites ($2R \approx 1$ nm)	Phase-separated state
Crossover $T_c$	$T_c \sim 10$ K (phase separation)	Melting of inhomogeneities

These scales determine localization, metallicity, and the observable superconducting properties at the interface (Zhang et al., 2018, Nanda et al., 2010).